MATRIX ANALYTIC METHODS APPLIED TO VARIOUS RISK PROCESSES
(Thesis format: monograph)
by
Kaiqi Yu
Graduate Program in Statistics
Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy
School of Graduate and Postdoctoral Studies The University of Western Ontario London, Ontario August 2008
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CERTIFICATE OF EXAMINATION
Supervisor Examiners
Dr. David A. Stanford Dr. Kristina Sendova
Co-Supervisor Dr. Bruce Jones
Dr. Jiandong Ren Dr. Henning Rasmussen
Supervisory Committee Dr. Qi-ming He
The thesis by Kaiqi Yu
entitled:
MATRIX ANALYTIC METHODS APPLIED TO VARIOUS RISK PROCESSES
is accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy
Date Chair of Thesis Examination Board ii ABSTRACT
Finding the ruin probability has been a popular topic in the literature of actuarial science. Since the finding of the connection between fluid flows and risk processes, the analysis technique of matrix analytic methods has been successfully applied to analyze a large class of risk processes. In this thesis, we continue along this line by applying matrix analytic methods to obtain ruin time related quantities arising in risk theory and the busy period in stochastic fluid flows as well.
The first issue of the thesis develops a unified approach for computing ruin probabilities for both infinite and finite time horizons for a large class of perturbed risk models. Specific members of this class include, among others, the perturbed
Sparre-Andersen model when distributions of inter-claim time and claim sizes are both phase-type, the perturbed Markov-modulated risk processes, and the per turbed risk processes with inter-claim times distributed according to a Markovian
Arrival process (MAP) and claim sizes being phase-type. From a ruin-theoretic point of view, the thesis establishes a unified, tractable phase-type structure for all variants of the class, and presents explicit formulas of manageable size for the ruin probability for models in this class. The key to the solution method is the ability to identify a sample-path-equivalent continuous-time Markov-modulated process with the risk process, and this is the sole limiting factor to the class of models that can be considered.
iii Whereas the analysis is exact in the case of the ultimate ruin probability, for finite time ruin probability, the Erlangization method is applied to the risk process to develop an approximation. An efficient recursive algorithm based on the block structure of the underlying generator matrices is given to perform the Erlangiza tion.
The second issue in the thesis provides a method to calculate the moments of the risk processes, which are the non-perturbed versions of the risk processes in the previous class. Based on the Laplace transform of the time of ruin, a recursive formula is established for the moments of the time of ruin by taking derivatives of the relevant Laplace transform. Numerical examples show that the initial procedure is unstable, although theoretically the result is valid. To overcome this problem, the uniformization technique is applied first to the underlying Markov process, which ensures that the algorithms obtained are stable and efficient.
Finally, we present similar procedures to computing the mean busy period of the perturbed fluid flow models.
For all cases, numerical examples are given throughout the thesis to illustrate the methodologies we developed.
Keywords: matrix analytic method, risk process, Markov-modulated risk model, continuous-time Markov-modulated process, time of ruin, ruin probability, fluid flow, Laplace-Stieltjes transform, moment, uniformization, busy period
IV CO-AUTHORSHIP STATEMENT
The materials presented in the thesis have been supervised by Dr. David A.
Stanford and Dr. Jiandong Ren.
Dr. Stanford and Dr. Ren proposed the study of the materials in Chapter
3. The goal is to obtain the ultimate ruin probability of perturbed risk models in a unified, tractable way in terms of the phase-type structures. My idea to ex ploit Asmussen [10] achieved this goal and we jointly determined the computation algorithm.
The idea of using Erlangization to approximate the finite time ruin probability in perturbed risk models in Chapter 4 was initiated by Dr. Stanford and we jointly formed the generator of the composite underlying Markov process for the perturbed risk model. Based on this idea, I developed the algorithm to approximate the finite time ruin probability. The algorithm is brute-force. Dr. Stanford and Dr.
Ren then suggested there is a repeating block structure for the generator of the composite underlying Markov chain. I verified this through numerical examples, and proposed the proof of it. I also derived recursive formulas for calculating the generator of the composite Markov chain.
Chapter 5 is related to the calculation of the moments of risk processes. Dr.
Ren and I jointly came up with the idea. The approach to calculating the moments of the time of ruin is based on the LST of the time of ruin. I found the way to
v obtain the moments by taking derivatives of the LST of the time of ruin (as given in Badescu et al. [16]) and derived the expressions of the moments in terms of the infinite matrix summation. Then I observed the algorithm was not stable in some situations. The idea of using uniformization to solve the stability problem belongs to Dr. Stanford. The algorithms for calculating the moments are mine. In addition,
I established theorems and gave the proofs in this chapter: for equations for ^(0) and M/(s) for s > 0; smoothness of the LST of the time of ruin; convergence of the algorithm; and so on.
In Chapter 5, we also considered the mean busy period of the perturbed fluid flow models. The idea to study this is mine. I derived the LST of the busy pe riod based on Dr. Ren's suggestion. The algorithm for the mean busy period and the proof of convergence of the algorithm belong to me. During the implementa tion, Dr. Ren suggested using reflection to obtain the correct probability of the upcrossing first passage time of the flow.
VI Dedicated to Jane, Sherry and my parents
Vll ACKNOWLEDGEMENTS
First of all, I want to thank my supervisors Dr. David A. Stanford and Dr. Jian- dong Ren whose help, stimulating suggestions and encouragement helped me in the time of research for and writing of this thesis. In particular, I would like to thank
Dr. Stanford for introducing me to the technique of matrix analytic methods. I sincerely thank him for all his advice over these years and I believe it will benefit me in my further work.
My sincere thanks to Dr. Qiming He, Dr. Bruce Jones, Dr. Kristina Sendova and Dr. Henning Rasmussen for acting as examiners.
Furthermore, I would like to express my heart-felt thanks to all the faculty, staff and graduate students in the Department of Statistical and Actuarial Sciences who have made my stay here both enjoyable and memorable.
I would never have reached this point without the selfless love, devotion and encouragement of my parents. Many thanks to my Mom and Dad for supporting me mentally so many years.
Above all, I am grateful to my wife Jane and my daughter Sherry for their
patience, support and companionship.
vm Contents
CERTIFICATE OF EXAMINATION ii
ABSTRACT iii
CO-AUTHORSHIP STATEMENT v
DEDICATION viii
ACKNOWLEDGEMENTS viii
TABLE OF CONTENTS xi
LIST OF TABLES xii
LIST OF FIGURES xiv
1 INTRODUCTION 1
2 MATHEMATICAL PRELIMINARIES 8
2.1 Notations and Conventions 8
2.2 Risk Processes 10
2.2.1 Definitions 10
2.2.2 Assumptions and Terminologies 14
ix 2.3 Matrix Analytic Methods 18
2.3.1 Discrete Time Quasi-Birth-and-Death Processes 18
2.3.2 Continuous Fluid Flow Processes 22
2.3.3 Matrices K and U 26
2.4 Risk Processes Analyzed as Markov Modulated Processes 30
2.5 Uniformization 34
3 RUIN PROBABILITIES IN PERTURBED RISK MODELS 37
3.1 Introduction 37
3.2 Mathematical Preliminaries 41
3.2.1 The Continuous-Time Markov-Modulated Process 41
3.2.2 The Perturbed Sparre—Andersen Variant 43
3.2.3 The Perturbed Markov-Modulated Risk Variant 43
3.2.4 The Perturbed MAP / PH Risk Variant 46
3.3 Pertinent Results from Asmussen's Fluid Flow Model 46
3.3.1 Determining U in the Pure Brownian Case 48
3.3.2 Determining U in the Partially Perturbed Case 50
3.4 Numerical Examples 53
4 ERLANGIAN APPROXIMATION TO FINITE TIME RUIN PROB
ABILITIES IN PERTURBED RISK MODELS 57
4.1 Introduction 57
4.2 Erlangization & Construction of the Composite Process 61
4.3 The Ruin Probability Prior to a Phase-Type Time Horizon 63
4.4 Simplifications for Erlang Horizons 66
4.4.1 The Structure of the Matrix XJH 67
x 4.4.2 Recursive Formulas for U^ ' and U)g ' 70
4.5 Examples 76
5 THE MOMENTS OF THE RUIN TIME IN MARKOVIAN RISK
MODELS 80
5.1 Introduction 80
5.2 Mathematical Preliminaries 85
5.2.1 A First Passage Time before an Exponential Horizon .... 87
5.3 Laplace-Stieljes Transform of the Time of Ruin 88
5.4 Moments of the Time of Ruin 92
5.4.1 Preliminaries 93
5.4.2 Computational Procedures 95
5.5 Convergence of the Algorithm 104
5.6 Examples 110
5.7 Mean Busy Period of Perturbed Fluid Flow Models 113
5.7.1 Laplace Transform of the Busy Period 113
5.7.2 The Mean Busy Period 117
5.8 Appendix 122
6 CONCLUSIONS AND FURTHER WORK 128
BIBLIGRAPHY 130
VITAE 139
XI List of Tables
3.1 Comparison of ruin probabilities 54
3.2 Parameters for contagion example 54
3.3 Comparison of ruin probabilities (a = 10~2) 55
3.4 Parameters used in the model 56
4.1 Comparison of the running time (T = 1000, u = 100) 76
4.2 Finite ruin time probabilities by Erlangian Approximation (t = 100,
u = 10) 77
4.3 Finite ruin probabilities by Erlangian Approximation (T = 1000,
u = 100) 79
5.1 Conditional standardized moments of the time of ruin Ill
5.2 Parameters used in the model 112
5.3 Conditional standardized moments of the time of ruin 112
5.4 Approximated conditional standardized mean of the time of ruin . . 113
xu List of Figures
2.1 Relationship between terminating Markov process {m(x)} and max
imum flow level M 25
2.2 Sample path of perturbed aggregate loss process and partially-
perturbed fluid flow 32
3.1 Coupling of aggregate loss and fluid flow processes 47
3.2 Decomposition of ruin probabilities varying according to a. Solid:
total ruin probability; Light solid: due to claim; Dashed: due to
diffusion 55
3.3 Decomposition of the ruin probability. Light solid: by our method;
Dashed: from Lu k Tsai [44] 56
4.1 The family of L-phase Erlangian distributions with common mean T. 60
4.2 Decomposition of ruin probabilities. Solid: total ruin probability;
Green solid: probability of ruin due to claims; Dashed: probability
of ruin due to diffusion 78
5.1 \E\ = 3. The three environmental states are denoted circle line,
solid line and dashed line 83
xm 5.2 One-to-one relationship of sample paths between {Vt}t>o and {Vt}t>o-
(red: the one crossing the bottom line is the reflected version) ... 116
xiv 1
Chapter 1
INTRODUCTION
Matrix analytic methods [45] provide a framework that is widely used to model and analyze, in a unified way and in an algorithmically tractable manner, a wide class of stochastic models, especially for queueing models. One successful use of matrix analytic methods is to analyze fluid flows by finding the stationary distribution function of the fluid level ([21] and [47]). At the same time, since the finding of the connection between fluid flows and risk processes [11], the analysis technique of fluid flows has been successfully applied to analyze a large class of risk processes
([2], [3], [16] and [48]). In this thesis, we continue this story of success by applying matrix analytic methods to solve problems arising in risk theory and stochastic fluid flows, particularly in the case of perturbed risk processes where they have not been used broadly.
The object of risk theory is to give a mathematical analysis of the random fluc tuation in an insurance business. Usually, risk theory in general and ruin theory in particular is considered as part of insurance mathematics. Ruin theory is a branch of actuarial science that studies an insurer's vulnerability to bankruptcy based on mathematical modelling of the insurer's surplus. It has been an active research 2 area from the days of Lundberg all the way up to today and many mathematical models and corresponding methodologies have been proposed.
The most interesting quantities in ruin theory are the likelihood of ruin and the time of its occurrence. In Lundberg's pioneering work, the Poisson process was used to model the inter-claim times, i.e., the time intervals between two consecutive claims are distributed as an exponential distribution with a common rate. The risk model with Poisson claim arrivals is referred to as the classical risk model.
In 1957, Sparre Andersen [59] generalized classical risk models by assuming that the inter-claim times are independently and identically distributed (i.i.d.) random variables and this generalized model bears his name. Since then, both the classical and Sparre-Andersen models have been used extensively to study ruin problems.
One extension of the risk models mentioned above is to assume that there is a perturbed component associated to the risk models. This perturbed component can be explained either as additional uncertainty of the aggregate claims or as an uncertainty to the premium income of insurer's surplus, and usually it is modeled as a Brownian motion.
One usually needs to study environmental impacts on the risk model (see As- mussen [12], page 146), and matrix analytic methods are frequently used to model a variable that influences the environment of the process. Thus, another variant of risk models was obtained by assuming that there is an independent process modulating the risk process in such a way that inter-claim rates and claim, sizes are influenced by this external process, which therefore allows more than renewal processes to be considered. Usually, the environmental process is assumed to be
Markovian. 3 Over the last twenty years, many authors have studied these risk models, and many methodologies have arisen to deal with them. We will review those which are related to our study. For a thorough survey, see Asmussen [12] and Rolski et al. [51].
For perturbed classical risk models, Dufresne & Gerber [27] derive convolution formulas for the probability of the time of ruin, which turns out to have compound geometric structures. Li Lu & Tsai [44] studied a perturbed equivalent of this model, where the variance of the Brownian motion is also influenced by the environmental process. They derived explicit formulas for the expected discounted penalty function of the ruin probabilities for the two-state model. For all three papers above, the target functions are derived by establishing and solving integro-differential equations. For the Li &; Garrido and Lu & Tsai papers, root finding techniques are necessary in order to obtain the target functions. The procedures are lengthy and not easily computable. In circumstances where the effort involved in obtaining the distribution of the time to ruin, which is quite cumbersome or even intractable, one alternative is to use the moments of this distribution in order to develop an approximation. There are several results for the moments of the time of ruin in the literature. In the classical risk model setting, Lin h Willmot [43] obtain the kth moment of the time of ruin in a recursive fashion with claim sizes being general i.i.d. random 4 variables. The starting point in their method is a result which gives the solution to a general renewal equation (see Lin & Willmot [42]); Drekic & Willmot [25] further give a simple expression for the kth moment of the time of ruin under the assumption that the claim amounts are exponentially distributed. They invert the Laplace transform (LT) of the time of ruin in order to obtain its distribution. Drekic, Stafford & Willmot [24] also consider a symbolic computation to combat the difficulties appearing in the computation of the moments by making use of the moment formulas given in [43]; Drekic & Willmot [26] further extend the result to the situation where the claim sizes are phase-type distributed. For Sparre Andersen model aspects, Dickson & Hipp [22] consider Erlang(2) risk processes as inter-claim time processes. Recently, Albrecher k. Boxma [8] study the Gerber-Shiu discounted penalty function in a semi-Markovian risk model. In both papers, the authors first derive Laplace—Stieltjes transforms (LST) of the time of ruin, and then obtain moments by taking derivatives of the LSTs. To get the kth moment, one needs to apply root finding methods. For the Albrecher &; Boxma formula, root finding needs to be used in each recursive step. From the computational point of view, it is not convenient. An initial motivation of this thesis is to alleviate some of the difficulties as sociated with finding ruin probabilities in perturbed risk models. To do so, we will make use of matrix analytic methods in a unified approach to calculating ruin probabilities for a large class of generalizations to the classical risk model. The original idea of using queueing models to analyze risk processes stems from Asmussen's work [11], where an M/G/l queue with jump service time is modulated by an environmental Markovian process. By constructing a sample- path-equivalent fluid flow representation of the Markov-modulated M/G/l queue, 5 the stationary distribution of the M/G/l queue is obtained in terms of the study of the corresponding fluid flow. It seems that Asmussen's work was the first to reveal the sample path equivalence between queueing models and risk processes. In another paper [10], Asmussen considers general fluid flow models with or without Brownian noise, and presents a complete methodology to deal with the stationary distributions of the fluid flows. Our thesis work draws heavily from this work. On the other hand, Ramaswami [47] analyzes the fluid flow queues in terms of matrix analytic methods. In his work, Ramaswami not only gives the repre sentation of the stationary distribution of the fluid flow level, but also presents a quadratically convergent algorithm to obtain it. In a follow-up paper to [47], da Silva Soares & Latouche [21] further extend Ramaswami's work by giving a more direct algorithm for the computation of the stationary distribution of the fluid level, as well as a clear probabilistic interpretation of that algorithm. The quantities determined in these two papers are useful to this thesis, in that all these quantities are tightly related to the determination of the stationary distribution of the fluid flow and turn out to be necessary for computing quantities related to the time of ruin and all have probabilistic interpretations as well. For a particular application of fluid flows and matrix analytic methods to risk processes, Badescu et al. [16] study the ruin probability of a fairly general risk model, i.e., these risk models have corresponding fluid flows. The technique used in [16] is an LST technique. Specifically, they first find the LST of the time of ruin, then invert it to obtain the finite time ruin probability. We should also point out that all their LST's are expressed in terms of the quantities given by da Silva Soares &; Latouche [21] or Ramaswami [47]. 6 One other quantity of interest in the fluid flow literature is the mean busy period of the fluid flow. The study of the busy period behavior dates back to the early work by Takacs [60] for queues. Recently, Asmussen [9] considered, among other things, the busy period r for fluid flow models. He obtained the explicit formulas for the mean of r in terms of the transition matrix and linear trends of the environmental process, and its LST by making use of the change of measure technique. More recently, in a series papers [5], [6] and [7], Ahn and Ramaswami analyzed the transient behavior of the fluid flow level by expressing the LST of the flow level in terms of the LST of the busy period of the fluid flow. The main focus of this thesis is to generalize the risk processes which we are able to convert into continuous-time Markov-modulated processes (see Definition 2.3.1), and further develop the use of matrix analytic methods to analyze them for the ruin time related quantities. The set of the risk processes we consider include Sparre-Andersen and Markov-modulated risk models. First, we are interested in determining the ruin probability in two contexts: the ultimate time ruin probabil ity and its finite time equivalent. The former case is based on Asmussen's work [10] and we develop a unified approach to calculating ruin probabilities for these models in Chapter 3. The latter case continues the first, and is based on the tech nique called Erlangization (see [13] and [55] for details). Erlangization gives an approximate solution for finite time ruin probabilities, and it can be improved by making use of an extrapolation method. In Chapter 4, we apply Erlangization to approximate the finite time ruin probabilities for the perturbed risk models consid ered in Chapter 3. In this chapter, we also develop an efficient recursive algorithm to obtain these probabilities. From a computational point of view, this algorithm saves computing time significantly. In Chapter 5, moments of the time of ruin of 7 the non-perturbed versions of the risk process applied in Chapters 3 and 4 are considered. We first establish theorems for the LST of the time of ruin, and then obtain recursive formulas for calculating those moments by taking derivatives of the LST. By resorting to the uniformization technique, the algorithms obtained are stable and efficient. Theoretical results are also given to support the algorithms. In addition, we present procedures similar to those for computing the mean busy period of perturbed fluid flow models in this chapter. Concluding remarks and further works can be found in Chapter 6. 8 Chapter 2 MATHEMATICAL PRELIMINARIES In this chapter, we introduce the necessary mathematical preliminaries, including notations and definitions of the models we study. A review of some of the existing results, methods pertaining to these models and quantities related to these models will be given as well, as many of these results, methods will be needed in the development of the results in the subsequent chapters. 2.1 Notations and Conventions The notations and conventions listed in this subsection are for later reference. Abbreviations c. d.f. denotes the cumulative distribution function of a random variable. i.i.d. denotes independent and identically distributed for random variables. w.r.t. denotes with respect to. CTMC denotes continuous-time Markov chain. DTMC denotes discrete-time Markov chain. LST denotes Laplace-Stieltjes transform. 9 LT denotes Laplace transform. Mathematical notations 8% denotes the set of real numbers. ^ denotes the set of complex numbers. JV denotes the set of non-negative integers. JV+ denotes the set of positive integers. P denotes a probability. E denotes an expectation. (•)T denotes the transpose of a matrix •. /IT denotes the kth moment of random variable r. I(^) denotes the indicator function of the event s/. • marks the end of a proof, an example or a remark, etc. Matrices and vectors All matrices and vectors are denoted by bold letters. Usually, matrices have up percase Roman or Greek letters like T, A, row vectors have lowercase Greek letters like a, 7v, and column vectors have lowercase Roman letters like t, a. In particular: I is the identity matrix, e is the column vector of appropriate length with all entries equal to 1, e, is the ith unit column vector, i.e., the ith entry is 1 and all other 0, [T]ij is the (i,j)th entry of the matrix T, [t]j is the ith component of t. ([t]i) is t. 0 is the null matrix (or vector), i.e., matrix (or vector) with all entries equal to 10 o, At and Aa represent block diagonal matrices with vectors t and a on the diagonals, respectively. ^ is the column vector with any ith component being reciprocal of [t],. jt| is the column vector with any ith component being absolute value of [t]j. 2.2 Risk Processes 2.2.1 Definitions The object of risk theory is to provide a mathematical analysis of the random fluctuation in an insurance business. Collective risk theory has been a popular tool to analyze the risk of an insurance company since Filip Lundberg investigated it in early of the last century. In collective risk theory, no particular attention is paid to individual policies issued by the insurance company; rather, the risk of the company is regarded as a whole. Two parties are involved in the process of mitigating the risk: the insurance company (insurer) on one side and all policy holders (insureds) on the other. The insurance company pays claims which are random events happening over the course of time, while the company receives risk premiums from the policyholders continuously or from time to time. In the present thesis, all risk models we consider are in continuous time. The aggregate claim amount that has occurred by time t is given by Nt th where {Nt}t>o denotes a counting process and Ui is the size of the i claim which 11 is a non-negative continuous random variable. The f/j's form a set of independent random variables. Premiums are received by the insurance company at constant rate c per unit time. The risk reserve process {Rt}t>o is then defined as Nt U 1 Rt = u + ct- Xt = u + ct~Y^ i' (2- ) where u is the initial surplus. The time of ruin is defined as T(U) = inf-0 > 0 : Rt < 0 | R0 = u} with T(U) = oo if ruin never occurs. The probability ip(u) of ultimate ruin is the probability that the reserve ever drops below zero: ip(u) =P(T(U) < oo) =p(miRt < o) . The probability of ruin by time T is Mu,T) = P(r(n) < T) = ^{^Rt < o) . For mathematical purposes, it is frequently more convenient to work with the aggregate loss process {St}t>o defined by Nt St = u-Rt = Y/Ul-ct. (2.2) i=i 12 Letting M = supj^t} and MT = sup {St} 4>0 0 I/>(U)=F(M>U), I/>(U,T) =F(MT>u) . The other quantity we shall be interested in is the moments of the time of ruin. The kth moment of T(U) is defined as k ^ = E{T(u) • I(T(U) < oo)]7 k = 0,1,2,... , where l(stf) denotes the indicator function of the event &/, which is equal to 1 if the event occurs, and 0 otherwise. The so-called classical risk model (2.1) is the compound Poisson risk model, where the counting process {Nt}t>o is a Poisson process. In the Sparre-Andersen setting, the counting process {Nt}t>0 is a general renewal process. In both cases, the claim sizes are i.i.d., non-negative random variables and all are independent of {iVt}t>0. Perturbed risk processes In Chapters 3 and 4, we will study ruin probabilities of perturbed risk processes. The diffusion component in the model is modelled as a Brownian motion and then 13 the perturbed aggregate loss risk process can be formed as St = 5]Zfc-ci + a&, (2.3) where a > 0, {£t}t>o is a standard Brownian motion. In this thesis, the perturbed component {aS,t}t>o should be interpreted as uncertainty in the premium income stream of the insurer. Because of diffusion, the probability of ruin can be decomposed as follows: i/)(u) = ipd(u) + ipc(u), where tpd(u) is the probability of ruin that is caused by oscillation or diffusion, i.e., that the surplus at the time of ruin is 0, and ipc(u) is the probability that ruin is caused by a claim, i.e., that the surplus at the time of ruin is negative. Mathematically, both can be expressed as ipd(u) = F[T(U) < oo, ST(U) = uj , l/jc(u) = F\T(U) < CO, ST(u) > UJ . Markov-modulated risk processes We may also let model parameters depend upon the state of an underlying continuous-time Markov chain. This is referred to as a "Markovian environment". One variant considered herein is the Markov-modulated risk process, whose inter- arrivals are not homogeneous in time, but determined by a underlying continuous- time Markov chain {Jt}t>o with finite state space E as follows. When Jt = i G E: • The arrival rate is /%, which means that during the time period when Jt = i, 14 the counting process is a Poisson process with rate /3,; • Claim sizes have distribution Bf, and • The premium rate is Cj. In addition, for perturbed components, the infinitesimal variance is of > 0. Under this setting, the aggregate loss process can be expressed as St = Y,Ui- [ cJvdv - [ ajMv, t > 0. (2.4) i=l J0 JO It is clear that when the underlying Markov chain has one state, the model will reduce to the classical risk model, that is, the arrival process is a Poisson process. 2.2.2 Assumptions and Terminologies All assumptions we make below apply throughout the thesis. We will point out exceptions whenever they occur. The continuous-time Markov chain We use an underlying continuous-time Markov chain (CTMC) to modulate another process. In this thesis, this underlying CTMC is of finite size and is irreducible and ergodic. Therefore, it has stationary distribution denoted as ir. The sub-intensity matrix In this thesis, a square matrix S with dimension £ x £ is called a sub-intensity matrix if [S]y > 0 (i ^ j) and ^?=1[S]jj < 0. A useful result regarding the distribution of eigenvalues of a sub-intensity matrix on a complex plane ^ is given 15 below (see Rolksi et al. [51]). Proposition 2.2.1 If S is a sub-intensity matrix, its eigenvalues lie in the left closed half of (if. The premium rate To assure that the ruin is not certain, all aggregate loss risk models considered in this thesis satisfy the following condition: S lim — = 7] < 0 w.p. 1, (2.5) t—>oo t which in turn requires a positive security loading. We first note that if there is a perturbed component, it does not affect the determination of the premium rate for the ruin probability being less than 1 since 1 " lim - / aJvd£v - 0 w.p. 1. For the Sparre-Andersen model, from Proposition 1.1 Chapter V of Asmussen [12], the condition (2.5) becomes cE[X] > E[U], where X and U are the common random variables for the inter-claim times and the claim sizes, respectively. For the Markov-modulated risk model, according to Proposition 1 in Jacobsen 16 [35], the condition (2.5) turns out to be ^^(ci-AE^]) >0, where Ui is the claim size random variable when environmental process in state i, (fti)ieE = 7r is the stationary distribution of {Jt}t>o- As noted in the Introduction, all risk models we consider in this thesis can be viewed as continuous-time Markov-modulated models (see Definition 2.3.1). Thus, all claim sizes are assumed to be phase-type distributed, which we define below. The phase-type distribution A phase-type distribution is defined as the time to absorption in a finite continuous-time Markov chain with one absorbing state. Precisely, we consider a Markov process on the states {0,1,..., m} with infinitesimal generator fo 0\ Q (2.6) Vt TJ where the m x m matrix T satisfies [T]** < 0, for 1 < i < m, and [T]^ > 0, for i 7^ j. Also Te +1 = 0, and the initial probability vector of the process is given by (ao, a), with a e + a0 = 1. We assume that the states 1, 2,..., m are all transient, so that, from any initial state, eventual absorbtion into state 0 is certain. A useful condition is given by the following theorem. Lemma 2.2.2 (see Neuts [45], P.43-45) The states 1,2, ...,m are transient if 17 and only if the matrix T is nonsingular. The above description leads to the following definition. Definition 2.2.1 (Neuts [45], P-45) A probability distribution F(-) on [0, oo) is a phase-type distribution if and only if it is the distribution of the time until ab- sorbtion in a finite Markov process of the type defined in (2.6). The pair (a, T) is called a representation of F(-). The basic analytical properties of the phase-type distribution are given by the following result. Recall that the matrix-exponential eK is defined by the standard series expansion X^Lo-^-"'/n^ where K is a square matrix. Proposition 2.2.3 (Asmussen [12]) Let B be phase-type with representation (a, T) and t= —Te. Then: (a) the c.d.fi of B is B(x) = 1 — aeTxe, x > 0; (b) the density function of B is b{x) = B (x) = oteTxt, x > 0; (c) the 1ST ofB is B[s] = a{sl- T)~H, s > 0; (d) the nth moment of B is $ - (-l)"n!are. One reason why we use phase-type distributions to model claim sizes stems from the fact that the family of phase-type distributions is dense in the set of all distributions on [0, oo). In principle, one can replace any non-phase-type distribu tion on [0, oo) by a suitable phase-type approximation (see Asmussen et al [14]). In addition, phase-type distributions are the logical matrix-based extension of the exponential distribution. In this regard (Asmussen [12], P.215) states "Phase-type distributions are the computational vehicle of much of modern applied probability. 18 Typically, if a problem can be solved explicitly when the relevant distributions are exponentials, then the problem may admit an algorithmic solution involving a reasonable degree of computational effort if one allows for the more general as sumption of phase-type structure, and not in other cases." The work in this thesis completely embodies this spirit, by converting the var ious risk models to continuous-time Markov-modulated models based on phase- type assumptions for claim sizes as well as other Markovian structures. There fore, analysis techniques of stochastic Markov-modulated models are crucial in our study and the matrix analytic method is one of the popular ones. 2.3 Matrix Analytic Methods The main purpose of studying queueing systems is to determine stationary dis tributions of quantities considered in these systems. The thinking behind matrix analytic methods is to deal with this problem in a unified way, so that we can focus on the algebraic systems, and avoid the other details not related to the problem. An example of this is the quasi-birth-and-death process, which is described next. 2.3.1 Discrete Time Quasi—Birth—and—Death Processes A common situation when we deal with M/G/1-type or G/M/1-type queues is to present the problem within the simpler framework of quasi-birth-and-death pro cesses (QBDs) (see, for instance, Neuts [45] and [46], and Latouche & Ramaswami [40]). These QBDs are Markov chains in two dimensions, the level and the phase, such that the process does not jump across several levels in one transition. Specifically, consider a discrete-time Markov chain {Xt}te_yy on the state space 19 U„>o^(n) where £(n) = {{n,j) : 0 < j < m} for n > 0. The first coordinate n is called the level, and the second one j is called the phase of the state (n,j). The word level also refers to the whole subset £(n). The number m of states in each level may be finite or infinite, but we confine ourselves to assume m is finite in our discussion. Such a Markov chain {Xt}te^r is called a QBD if, in a single transition, the chain can move upwards only to the next higher level, downwards only to the next lower level, or remains at the same level. In the following, we assume that {Xt}t^^y satisfies such conditions and therefore {Xt}te^r is a QBD. If the transition probabilities of {Xt}tEt/r are level-independent, i.e., a single transition probability does not depend on the current level of the chain, then, the transition probability matrix of {Xt}teyy is block-tridiagonal and has the following structure: / \ B An 0 0 A2 Aj A0 0 '•• Q = 0 A2 Ai A0 '•• 0 0 A2 Ai v : •• •• •• •/ where Ao, Ai, A2 and B are square matrices of order m. If further {Xt}te^ is irreducible and ergodic, and its stationary distribution vector is denoted by 7r, and if furthermore we partition 7r by levels into subvectors Tfn, n > 0, where 7r„ has m components, a remarkable result for this QBD (see 20 M.F. Neuts [45]) is that the solution 7r may be written as ivn = vi-oR™ for n > 0, (2.7) where R is an m x m matrix, which is called the rate matrix in the literature. R can be interpreted as follows: for any n > 0, [R]y- (1 < i,j < m) is the expected number of visits to (n + 1, j) before a return to 1(0) U£(l) U • • • L)£(n), given that the process starts in (n,i). As pointed out by Latouche & Ramaswami in [40], the matrix-geometric prop erty (2.7) characterizes the stationary distribution 7r and also leads to simple ex pressions for other quantities, like marginal distributions and moments of the phase and the level, but it is not suitable for numerical evaluation of R (see Page 134- 136 of [40]). Because of this, two other matrices U and G come into use. All three matrices R, U and G play important roles in the study of QBDs both from theoretical and computational points of view. The matrix U is the probability matrix, starting from level £(n), of returning to £(n) before visiting £(n — l) and the matrix G is the probability matrix, starting from level £(n), of ever visiting £{n — 1). Due to the homogeneity of the matrix Q, the values of U and G do not depend on the level n > 1. The relations among R, U and G are listed below (see Latouche & Ramaswami [40]). All these can be obtained by conditioning arguments. 21 R = A0(I-U)-\ (2.8) G = (I-U^Aa, (2.9) U = Ai + AoG, (2.10) = Ai + RA2. (2.11) Thus, if any one of these three matrices is known, then we may determine the other two by applying the relevant equations above, and to determine one of them, the following non-linear matrix equations for R, U and G can be applied: U = Ai + Aotl-U)-^, (2.12) G = Aa + AiG + AoG2, (2.13) 2 R = A0 + RAi + R A2. (2.14) All these matrix equations are not only useful for computational purposes, but also possess probabilistic interpretations. For example, for Equation (2.14), we know that the matrix R records the expected number of visits to £(n), starting from £{n — 1), before the first return to £{n — 1). In the right-hand side of (2.14), these visits to £{n) are decomposed into three groups according to the level from which a visit to £{n) occurs: (1) From the level £{n — 1), which means that it is the first visit to the level £(n) with probabilities given by AQ; (2) From the level £(n), which means that each visit to £{ri) with expected number R is immediately followed by another visit to £{n) with probabilities given by Ai; (3) From the level £(n + 1), which means that each visit to £{n + 1) with expected number R2 is 22 immediately followed by a visit to £(n) with probabilities given by A2. Although we may first compute any one of R, U and G for the purpose of determining the stationary distribution, working on matrix G has the additional benefit that G is necessarily stochastic, i.e., Ge = e. In contrast, R is not stochas tic and U typically is not. Iteration algorithms are the easiest ways to obtain G based on Equation (2.13). One typically exits the iterative loop when each com ponent of |Ge —e| is less than prespecified tolerance. See Latouche & Ramaswami [40], Chapter 8, for details. Another attractive reason to work in QBD framework is that there is a quadrat- ically convergent algorithm, call the Logarithmic Reduction (LR) Algorithm ([40]), to solve Equation (2.13) for G. Thus, determining the stationary distribution iz becomes available and efficient. 2.3.2 Continuous Fluid Flow Processes The classical results for QBDs pertain to models in which the level is a discrete quantity, which changes in unit jumps. About a decade ago, some researchers investigated stochastic models in which the level would be a continuous quantity, increasing or decreasing linearly over time. Space does not permit us to review the entire literature, but here we review the papers by Asmussen [10], Ramaswami [47] and de Silva Soares & Latouche [21] on determining the stationary distribution of the fluid level of a continuous-time Markov-modulated fluid flow. The model setting is as follows. A fluid level process {V(}t>o with infinite fluid buffer with input rates are modulated by a continuous-time Markov chain {Jt}t>o- The initial level is assumed to be 0. The state space E of {Jt}t>o is partitioned into two subsets E+ and E~: when state i e E+, the fluid buffer increases with rate 23 rt > 0, and decreases with rate r7 < 0 while the fluid level is nonempty and the state j G E~. The infinitesimal generator A of {Jt}t>o is accordingly partitioned as / A(++) A(+-) \ (2.15) y A<"+> A(~) y The necessary and sufficient condition for the existence of the stationary dis tribution of {Vt}t>o is that the average drift of the flow is negative, that is (see Asmussen [10]), Y^ TTJ n < 0 , where n = (7^) is the stationary distribution of {Jt}t>o and will be partitioned as (7r(+),7r(~)) according to ascending and decreasing phase sets of {Jt}t>o- This assumption is assumed throughout the thesis. One other quantity related to the fluid model is the net input to the buffer which is defined as, up to time t, St= rJvdv. Jo {St}t>o is a stochastic process and the fluid level process {Vt}t>o is then Vt = St- min Sv 0 Since the process {(Jt, St)}t>o is the main objective we will study in this thesis, we give the following definition. Definition 2.3.1 The stochastic process {(Jt,St)}t>o defined above is called a 24 continuous-time Markov-modulated process (CTMMP). / r<+, A Denote r = as the vector of the linear rates of the fluid flow process r(-> {Vt}t>o, where r^ and r^~) are the vectors of the rates when the process {Jt}t>o are in E+ and in E~ respectively, and define A<++) A(+-> kVkl • A<-+> A<-> V / (+_) ' T(++) T (2.16) T(-+) T(—) V ; The essential use of the matrix T is to change rate per unit time to rate per unit fluid. According to Asmussen [10], this operation is named as an operational time argument. The characteristic of the stationary distribution of the flow level process {Ft}t>o was first given by Asmussen [10]. The ideas and methods used in [10] are very intuitive and can be accommodated to various situations. In this thesis, we will apply these ideas and methods to some risk models to obtain quantities related to the time of ruin of the risk model. Consider fluid flow process {{Jt,Vt)}t>o with negative average drift, and its , net input process {S t}t>o. To determine the stationary distribution of the flow level, Asmussen first found that the problem of computing this distribution is equivalent to finding the distribution of the maximum of the time-reversed version of {(Jt, St)}t>Q. On the other hand, we are interested in the distribution of the maximum of the continuous-time Markov-modulated process. Thus, Asmussen's 25 method can be directly applied for our purpose. Since the average drift of the flow is negative, the maximum of {5t}t>o> M = maxt>0St, is an almost surely finite random variable. For all non-negative levels x < M, Asmussen defines m(x) to be the state of the underlying Markov chain {Jt}t>o at the first instance that the process {St}t>o reaches level x. Obviously, the values of {m(x)} are in E+. Figure 2.1 (redrawn from Asmussen's Figure 1 in [10]) is a typical example, where there are two increasing phases denoted as 1 and 2. By sample path observation, Asmussen establishes the following result: o LO o o CO o CN O O O Figure 2.1: Relationship between terminating Markov process {m(x)} and maxi mum flow level M Proposition 2.3.1 {m(x)} is a terminating Markov jump process with life-length coinciding with M. 26 Since the irreducibility and finiteness of the underlying Markov chain, all non- absorbing states of this phase distribution are transient. Then, by Lemma 2.2.2, we conclude that Proposition 2.3.2 The maximal value M of {S}t>o is phase-type distributed and the phase generator of it is non-singular. Besides the characteristics of the stationary distributions for fluid flow models, perturbed fluid flows were also considered in [10], so that Proposition 2.3.1 is applied equally for perturbed cases. 2.3.3 Matrices K and U Ramaswami [47] also shows that the stationary probability density TT(X) of the fluid level process {Vt}t>0 has a matrix exponential form by a ladder-height argument, and proves that the distribution is phase-type by making use of the dual process of {(Jt, Vt)}t>o- Then last but not least, Ramaswami constructs a very efficient algorithm based on the LR algorithm to compute the steady state distribution ir(x) by reducing the analysis of the computation for a continuous-time, continuous- state space system to a related QBD framework, which is a simpler discrete-time, discrete-state space system. Partition TV(X) in a manner conformant to TV as ir(x) = (7r(+\x), 7r^\x)), then Ramaswami obtains the following theorem: Theorem 2.3.3 (Ramaswami, [47]) There exists a matrix K of order \E+\ x \E+\ and a matrix ty of order \E+\ x \E~\ such that the steady state density vector of the fluid flow model {(Jt, Vt)}t>o *s of the matrix exponential form (7T+(x),7v~{x)^a[eKx,eKx^, x>0, 27 th Kx where ex = —TT+K. The (i,j) element of the matrix e records the expected number of visits to (x,j), j G E+ before returning to level 0 given that the process starts in (0,i), i G E+. The (i,j)th element of the matrix eKxiS records the expected number of visits to (x,j), j G E~ before returning to level 0 given that the process starts in (0, i), i G E+. From the preceding theorem, we know that the matrix K is the intensity matrix of the process {Jt}t>o embedded at visits to phases in E+. As pointed out by Ramaswami, K is nonsingular (due to the negativity of the average drift), and can be expressed as K = T(++) + * T(~+) . Furthermore, /•OO q,= / e^T^e^-^dy. Jo One interpretation of \1/ based on it is as follows. Given a level y > 0 and the flow starts in (0, E+) at time 0, conditioning on the last change of the phase from E+ to E~ at level y before the level returns to 0, since eKy is the average number of visits to state (y, E+) and eT v is the probability of the flow to level 0 from y, then [vE^y, i G E+ and j G E~, is the probability that, starting from (0, i) at time 0, the fluid flow returns to the level 0 at some finite time at phase j. In a follow-up paper to [47], da Silva Soares &; Latoucthe [21] extend Ra- maswami's work by expressing a more direct algorithm that avoids the use of the dual process of {(Jt, Vt)}t>o for the computation of the stationary distribution of the flow level, as well as a clear probabilistic interpretation for their algorithm. 28 Among these results, under the same model setting as Ramaswami used, da Silva Soares & Latoucthe obtained the following result. Theorem 2.3.4 (da Silva Soares & Latoucthe, [21]) The matrix M/ satisfies /»oo ++) *= / e^ ^I<+-)e^dy, (2.17) Jo where U= T^ + T"+*. Based on the expression (2.17), another interpretation for Sfr can be given as follows. Conditioning on the first transition from E+ to E~ at rate T^"^ and at some level y > 0, the probability of the flow down to the level 0 from y is eUy. Since T^++) is the transition matrix of the flow remaining in phase set E+, the probability of flow returning to level 0, starting in (0, E+) is given by (2.17). Clearly, for i, j € E~, [eUx]ij is the first passage probability from the level x > 0 to the level 0, starting from i and ending at j within a finite time period. Mathematically, if 6 is the time instant of the flow first reaching the level 0, Vx [e ]zj = v(e < co, J0=j\Vo = x,Jo = iy Another contribution of [47], which is derived in a different way in [21], is that the authors construct a direct algorithm for the stationary distribution by convert ing the computation of \& to the discrete-time QBD framework. The procedure is as follows. First, uniformize the phase process {Jt}t>o by defining P = I + -T, where H > maxje£:{—[T]jj}, and decompose matrix P in a manner conformant to the 29 partition of T as ' p(++) p(+-) » (2.18) p(-+) p(—) Then, construct a QBD so that *$? can be interpreted as the first passage prob ability matrix from states (1,E+) to states (0, E~) for this QBD. Next, define transition blocks, each has conformant decomposition to that of P, (hi o "\ ( |p(++) o \ / o lp(+-) ^ An and A2 . (2.19) 0 0 p<-+) 0 o p<~> V \ ; V J Last, determine the G matrix for this QBD, and since G represents the matrix of first passage probabilities to lower levels for the QBD process defined by (2.19), it follows that the ^ matrix occupies the northeast corner of G; that is ' 0 * * G = (2.20) \° ') where matrix block * represents the elements of G which we are not concerned with. Remark 2.3.1 We will see that the matrix \I/ plays the most important role in our study, and one other expression for \& will be given in a later chapter, which turns out to provide us a computable procedure to obtain it. At this stage, it suffices to observe that ^ bridges the two fundamental matrices K and U. Due to this 30 importance, we repeat these relations here: K = I<++) + * 1<-+), U = I<~} + 2<-+) * , ' e^l^e^^dy, 0 e^yj(+-)euydy. 2.4 Risk Processes Analyzed as Markov Modu lated Processes Throughout this thesis, matrix analytic methods are techniques used to analyze risk processes. The construction of the sample-path-equivalent continuous-time Markov-modulated process to the risk process is the first step in our work. We introduce the construction in what follows. To that end, we start with considering a continuous-time Markov chain {Jt}t>o which modulates the inter-claim time and the claim sizes of the aggregate loss model defined by (2.3). The states of this process are grouped into two subsets: the set E+ of phases for the claim sizes and E~ for the inter-claim time. The wm infinitesimal generator A of {Jt}t>o accordingly be partitioned as /A(++) A(+-)\ A = (2.21) yA(-+) A(-)y We call E+ the set of ascending phases and E the set of descending phases. The evolution in time of the aggregate loss process {St}t>o can be related to the 31 evolution of a continuous-time Markov-modulated process (CTMMP) in terms of {Jt}t>o- The associated CTMMP is denoted as {(Jt, St)}t>o, where the value of St at time t will be clear soon. Specifically, there is a one-to-one relationship between the sample paths of {S't}t>o and {St}t>o- The constructed CTMMP has rates of increase and decrease based on the underlying states of the Markov chain {Jt}t>o- For risk models described by (2.3) that can be accommodated in this way, claim payments correspond to the evolution of the increasing periods and inter-claim time to the decreasing ones with perturbation. The decreasing rate of {St}t>o during the decreasing phase i € E~ of {Jt}t>o is same as the premium rate when Jt = i. Since the payment is paid at an instant in time, the increasing time periods for {St}t>o are "artificial" and we choose all corresponding increasing rates to be 1 throughout this thesis. Figure 2.2 shows a typical sample path of {St}t>o and that of the corresponding risk process {St}t>o- Having made this construction, the study of the ruin problem of the risk model will reduce to the study of the corresponding CTMMP for the problem of the first passage time. We observe that the approach to constructing a sample-path-equivalent CT MMP from a risk process defined by (2.3) is also applicable to other variants of the model, for instance, those defined by (2.2), which is a non-perturbed risk model. This is because only the counting process and the claim size distributions are considered in construction. As illustration, we give two examples. The (perturbed) Sparre-Andersen model For the Sparre-Andersen model, we assume that the premium rate c and in finitesimal variance of the Brownian motion a (if perturbed) are constant, and 32 Figure 2.2: Sample path of perturbed aggregate loss process and partially- perturbed fluid flow the common distribution of the inter-claim time is phase-type with representa tion (a, S) with state space Es and the common distribution of the claim sizes is phase-type with representation (/3, G) with state space EG. The environmental process {Jt}t>o is constructed as follows. Its state space E consists of two disjoint sets: The ascending phase set corresponding to the claim size phases, i.e., E+ = EG and the descending phase set corresponding to the inter- s claim phases, E~ = E . Then, the resulting infinitesimal generator of {Jt}t>o is given by ' G go: * ' A(++) A(+-) \ A = (2.22) \s/3 S J A(-+) A(-) ; where s -Se and g = —Ge. The (perturbed) Markov-modulated risk model 33 We consider a Poisson risk process in a Markovian environment with the nota tion of Section 2.2.1. That is, the underlying Markov chain with state space E is {Jt}t>o, and the intensity matrix is A. The arrival rate while the underlying pro cess is in state i is $ and the size of an arrival claim is Bi (a random variable). We assume that each Bi follows a phase-type distribution with representation (a,, GJ) with state space E^\ In order to get the sample-path-equivalent CTMMP of this Markov-modulated risk model, we first convert the risk process to the corresponding CTMMP for each background state i, and then combine the Markov environmental process with each converted CTMMP to keep track of phases of the claim sizes and the background process. We denote this combined phase process as {J}t>o, which is constructed as follows (Asmussen [12], Page 234-236). The state space E of {J}t>o is defined as E = EU{{i,a) :ieE,a / A A A \ A = (2.23) V Afc) A(G.) J where gj — —G^e, and A^), A^^), A(gi) and A(G,) are block matrices with corresponding matrices on the diagonal. 34 2.5 Uniformization There are two main study issues under the umbrella of matrix analytic methods: One is the matrix-geometric solutions for discrete systems (see Neuts [45]). A particular case of this is the solution of the stationary distribution given in (2.7) for QBDs. The other is matrix-exponential solutions for continuous-time sys tems. The exponential expression for the stationary distribution of the fluid flow level given in Theorem 2.3.3 is one example for this case. One other example is the calculation of the probability transition matrix for a continuous-time Markov chain. Both methods provide us theoretical formulas to employ to compute the sta tionary distribution and the probability transition matrix mentioned above. Com puting the stationary distribution for QBDs is straightforward, and efficient due to the LR algorithm. For matrix-exponential solutions, there exist several methods in various situations. For example, by making use of differential equation tech niques, one can get the probability transition matrix P(£) for a continuous-time Markov chain. In the following, an alternative method to get P(i), based on a probabilistic construction, is presented. The technique is called uniformization (for example, see Ross [52]). In fact, in Section 2.3, this technique has already been used to construct a QBD to obtain the matrix \I/ via (2.20). First, consider a CTMC {Xt}t>0 with exponential rate in each state being a common constant //. Its infinitesimal generator is denoted as Q. We denote by P the probability transition matrix associated with the DTMC embedded at transition epochs. By conditioning on the number of transitions n to occur in time 35 t, the probability transition matrix of {Xt}t>0 can be shown to be given by P(t) = eQ* = ^P«e-M*M_, t>o. (2.24) 71=0 The last equality in Equation (2.24) is useful from a computational point of view since P™, for all n, are probability matrices, which means that all entries in Pn are non-negative and their sum does not exceed 1. Thus, it enables us to approximate P(t) by taking a partial sum and then computing the relevant n-stage probability matrix P™. The procedure is stable. Whereas the CTMC for {Xt}t>0 above is limited in practice, it turns out that the CTMCs with rates for states being unequal can be put in that form by allowing fictitious transitions from a state to itself. Specifically, assume {Xt}t>0 is of this more general form (i.e., with unequal rates), with infinitesimal generator Q. Denote fa = — [Q],, and assume that {fa} are bounded, and define \x such that /-* > M», for all i. Define P = I + -Q, (2.25) A4 then, for t > 0, nc-UtW £p n! n=0 oo n / \ ^ , _ . e n! n=0 fc=0 = eQt 36 which is the transition matrix of the process {Xt}t>o- Therefore, we may use the method descried for Equation (2.24) to obtain probability transition matrix e^* of n e lit {Xt}t>Q through expression Y^=o^ ~ ^~- Thus, from a computational point of view, uniformization provides us with a convenient way to compute the transition matrix P(t) for a CTMC without resorting to solving differential equations. The probabilistic interpretation of uniformization is as follows. When in state i, the process {Xt}t>o leaves at rate \x. However, only with probability yUj/yU does the process actually leave state i, whereas with probability 1 — fii/fj,, the process remains in it. In Chapter 5, the uniformization technique will be used to handle the stability issue when we deal with infinite matrix summation to obtain moments of the time of ruin of risk models. 37 Chapter 3 RUIN PROBABILITIES IN PERTURBED RISK MODELS 3.1 Introduction In this chapter, we present explicit formulas for calculating ruin probabilities for a large class of generalizations to the classical risk model. The analysis of the time of ruin will switch to the study of the first passage time of the sample-path-equivalent CTMMP of the risk process. Thus, the methods developed in this chapter for the ruin probability are applicable to all risk models which have corresponding CTMMP equivalents. This class of risk models includes quite common risk models in the literature of actuarial sciences such as the perturbed Sparre-Andersen model when the distributions of the inter-claim time and the claim sizes are both phase- type, and perturbed Markov-modulated risk models such as those studied by Lu & Tsai [44]. The primary contributions of this chapter are two-fold. First of all, we provide an answer to the question as to how large a class of processes has a phase-type structure for the aggregate loss process. Secondly, this chapter provides a general method to calculate the ruin probability for a class of quite general perturbed risk 38 processes and it turns out to present a unified algorithmic approach or "template" for the computation of the ruin probability for all members of that class. Because of the phase-type structure, the algorithm for calculating ruin probabilities is stable and readily programmable. The generic stochastic process considered in this chapter is the following ag gregate loss process: Nt St = '^2zk-ct- a£u (3.1) fc=i where the process {A^}t>0 counts the number of claims up to time t and {£t}t>o is a standard Brownian motion. Let T(U) = inf{£ > 0 : St > u} denote the time of ruin, where u is an initial surplus of the insurance company, and M = supt>0{5t} the maximal aggregate loss. The ruin probability is given as ip{u) = F(T{U) < oo) =P(M > u), u>0. (3.2) The probabilities of ruin due to diffusion and due to claims are denoted by Mu) = ^(T(u) < °°; Sr(u) = u) (3.3) and ^c(u) = F(r(u) < oo; ST(u) > u), (3.4) respectively. Obviously, ip(u) = ipd{u) +4>c{u). In Dufresne & Gerber [27], it was established for the perturbed classical risk pro cess that the maximum aggregate loss could be expressed as a compound geometric distribution (see their equation (3.4)). Each term in this sum is the convolution 39 of an exponential amount for the drift occurring prior to the ladder height due to the claim and an equilibrium claim-size distribution for the ladder height due to the claim itself, plus one convolution with the final amount that occurs due to the drift occurring after the final ladder height. When the claim amounts themselves are phase-type distributed, this compound-geometric structure turns out to be phase-type as well. This is because the equilibrium distribution of a phase-type distribution remains phase-type, and the convolution of phase—type distributions is again phase-type. Thus, computing procedures for the ruin probability are available through the distribution of the maximum aggregate loss. Li & Garrido [41] considered a Sparre—Andersen model in which the inter-claim times followed a generalized Erlang distribution of order n, from which it can be seen that the contribution to the ladder height due to the diffusion comprised a linear combination of n exponential terms. This hinted at the possibility that the contribution due to diffusion in the presence of phase-type inter-claim times might be phase-type, in which case the resulting maximal aggregate loss would likewise be phase-type. Another member of the class we seek to discern is the so called Markov- modulated risk process, where the distributions of the claim frequencies and sizes are influenced by an independent Markovian environmental process. Quite re cently, Lu & Tsai [44] studied a perturbed equivalent of this model, where the variance of the Brownian motion is also influenced by the environment process. By establishing and solving integro-differential equations, they derived formulas for the expected discounted penalty function for the two-state model. In their pa per, they used general positive random variables as claim sizes and so there is no phase-type structure. But in their numerical example, Erlang(2) and a mixture of 40 two exponential random variables were used as claim sizes, and the expressions of the ruin probabilities are expressed as a combination of two exponential functions and their coefficients are trigonometric functions. We suspect that the resulting distribution of the aggregate loss actually has phase-type structure. Our results show that it is true. It is therefore of natural interest to try to discern the possible class of models that would possess similar phase-type structural results, and the common charac teristics for that class. It is of equal interest to us to determine efficient compu tational procedures (i.e., a sort of computational "template") for all members in that class. Once the parallel CTMMP is identified, we may employ the method ology from Asmussen [10], which concerned the distribution of the maximum of a CTMMP. This approach, presented in Section 3.3, gives rise to a tractable, easily comprehensible algorithm involving matrices of manageable size for the maximal aggregate loss, and hence the ruin probability. In addition, the formulas developed all have probabilistic interpretations. Finally, Section 3.4 presents a range of numerical examples that illustrate the breadth of models that can be considered. We should point out that the phase-type class is not particularly restrictive: almost all standard examples in the ruin theory literature we have encountered can be represented easily by this class (including generalized Erlang distributions and mixtures of Erlang distributions) due to the fact that any positive random variable can be approximated by phase-type distributions. Heavy-tailed distributions can not be represented exactly by the phase-type class, but as with many other classes of fitting distributions, they can be approximated by the phase—type class to any desired degree of accuracy (for instance, see Thorin & Wikstad [61] and [62]). In 41 addition, an EM algorithm (for example, Asmussen et al. [14]) is also available for fitting the distribution. Overall, this chapter relies heavily on previous work by Asmussen [10]. How ever, we appear to be the first to recognize the common structure of the class that gives rise to ruin probabilities of a phase-type form. Furthermore, we present a unified approach for the solution of all problems belonging to this rather large class. 3.2 Mathematical Preliminaries In the remainder of the chapter, we consider the risk models defined by (3.1) which have corresponding sample-path equivalent continuous-time Markov-modulated processes (CTMMPs). Although the construction of this sample-path equivalent from a risk process was introduced in Chapter 2, we give a brief description here to emphasize it. 3.2.1 The Continuous—Time Markov—Modulated Process As introduced by Asmussen [11], given any sample path for a risk process (see sample path in the top half of Figure 2.2), a sample-path-equivalent CTMMP (the lower part of Figure 2.2) can be constructed as follows. (The methodology works equally well for perturbed as for non-perturbed risk processes.) During periods between claims, the sample paths are identical. To represent the jumps in the risk process that correspond to claims, we introduce artificial time segments during which the process increases linearly with slope 1. By doing so, the amount of elapsed "artificial" time equals the claim size, as does the amount of rise. 42 When the inter-claim times and claim sizes have a Markovian structure, we may associate the aggregate loss process {St}t>o with a sample-path-equivalent CTMMP {(Jt, St)}t>o- The process {Jt}t>o keeps track of the underlying phases of the Markov chain which affects the rate of the evolution of {St}t>o at time t. We can relate the aggregate loss St of the risk process to the associated process St + via the relation St = St+T+, where T is the sum of the artificial time segments up to time t. In other words, we can fully characterize the aggregate loss in terms of the associated CTMMP. We start by grouping the states of the Markov process {Jt}t>o into two sets: + a the set E" for inter-claim times, and the set E for claim sizes. While Jt = i G. E ', the aggregate loss process behaves like a Brownian motion with a negative drift Q + and infinitesimal variance of. While Jt — i € E , there is no passage of time in the risk process; however, the associated CTMMP simply increases at unit rate, not subject to influence from the Brownian motion. The generator of the phase process of {Jt}t>o is A = (3.5) A<+*> A(++) J (We have used the subscript a to highlight the fact that the influence of the Brownian motion is limited to the inter-claim times of the original risk process.) We assume that {Jt}t>o is irreducible and has limiting distribution TV = (7^). In what follows, we illustrate the wide range of applicability through several examples. 43 3.2.2 The Perturbed Sparre-Andersen Variant This mapping of a risk process to a CTMMP is possible for the perturbed Sparre- Andersen models with both the inter-claim times and the claim sizes having phase-type distributions. Specifically, let the distribution of the inter-claim times be phase-type with representation (S,S) with state space E" — {1,2, ...,m} and the distribution of the inter-claim times, and claim sizes have representa tion PH(/3, G) distribution with state space E+ = {1 + m, 2 + m,..., n + m}. Then the linear drifts and infinitesimal variances of the process {St}t>o are vectors r = { —c, —c,..., —c, 1,1,..., 1} and a2 = {a2, a2,..., a2}, where —c is the drift a + 2 when Jt G E and 1 is the drift when Jt G E . There are m —c's and cr 's, and n l's above. The infinitesimal generator of the underlying Markov process {Jt}t>o is given by (ff+) A(<™) A | | S sj9 A = A(+*) A(++) I 3.2.3 The Perturbed Markov-Modulated Risk Variant In the context of Markov modulated risk processes, we set the risk process {54}t>o defined in (3.1) in a Markovian environment and let the premium rate, the in finitesimal variance of the Brownian motion, and the parameters of the claim amount distribution be governed by an independent Markov process {It}t>o with finite state set E® = {1,2,... ,n} and initial distribution 7. Its intensity matrix is denoted by Q which is assumed to be irreducible. When It = i, the premium rate is c,, the infinitesimal variance of the Brownian motion is of, the arrival rate is Aj and the distribution of the claim size is Bt which is phase-type distributed with representation (aW,T") and state spaces 44 w (i) E(i) = j-1») 2«)...; gW}. We denote by t = -T e the absorption rates of the phase process associated with Z?j. In this case, the phase set of the underlying Markov process {Jt}t>o is E = Ea U E+, where Ea = EQ and E+ = UiE®. The vector of linear drifts of the flow {St}t>0 is r = {rn,rp} = {-cu -c2,..., -c„, 1,1,..., 1} , where rn = a {—ci, — C2, •.., — c„} is the vector of drifts in states E and rp = {1,1,..., 1} is the vector of drifts in states E+. The vector of infinitesimal variances associated with the perturbation component in states Ea is denoted by cr2 = {a2, a2,..., cr2 }. With this setup, the infinitesimal generator of the underlying Markov jump process {Jt}t>0 is given by Av'"> A^ " Q ~ A(A») A(A,c«W) A (+CT) A (++) A(t(0) A(T(i)) J where Awijs, A(AiCe{ij) and A,-t(,)) denote block diagonal matrices with elements T^, \a^ and t^ on the diagonals. We note that this generator is actually derived in Asmussen [12], page 234-236, where a mapping between the unperturbed risk process and the the fluid process was introduced. As pointed out in Chapter 2, the perturbation does not change the underlying Markovian structure at all. A Model of Contagion To illustrate this type of model further, we resort to an example first presented in Badescu et al. [16], which assumes that an underlying two-state Markovian environment controls the claim rates and claim sizes. There is a predominant normal state, A and a "rare" state B to represent periods of contagion. The system switches from A to B at rate a A and from B to A at rate as- Environment A 45 features standard claim rates and claim sizes, while environmental B also features a supplement stream of claims due to a highly infectious disease. Claims occur in two ways. There is at all times a Poisson process with param eter 5\ of small exponential claims, their mean being —. While in environment B, there is in addition a second process, with parameter S2 of exponential claims; their expected value is —. The following five states are identified: 1. environment A, during an interval between claims; 2. environment B, during an interval between claims; 3. environment A, normal claim payment in progress; 4. environment B, normal claim payment in progress; 5. environment B, contagion claim payment in progress. From the expression (3.6), the generator is given by —aA - <5i OLA Sx 0 0 aB -aB — Si — &2 0 5\ 52 Hi -m o o Mi 0 -MI 0 M2 0 0 -ii2 46 3.2.4 The Perturbed MAP / PH Risk Variant One can further generalize the previous case by assuming that the risk process' underlying state evolves over time according to a Markovian arrival process with representation MAP(5+,D0,Di). Essentially, [D0]y for i ^ j; i,j & {l,...,m} contains the rate of change from underlying state i to j without an associated claim occurrence. [Di}ij for i,j € {1,..., m} represents the rate of change from underlying state i to j accompanied by a claim occurrence. S+ denotes the starting distribution on the underlying states {1,... , m}. Whenever claims occur, an independent claim amount from a common phase type distribution with representation (/3, G) of order n occurs. In this way, it is possible to induce a correlation among successive inter-claim times while keeping the claim size process independent. In the associated CTMMP, this is achieved by "freezing" the underlying inter- claim state while a claim is paid. Thus Ea = {1,..., m} and E+ = {1,..., n} x {!,..., m}, so that D0 /3®Di A (3.6) y g®I G(g)I J 3.3 Pertinent Results from Asmussen's Fluid Flow Model As mentioned in the Introduction, in this chapter, we consider aggregate loss pro cesses given by (3.1) which can be associated with sample-path-equivalent CT- MMPs. Then the study of the maximum of the aggregate loss process {St}t>o can be reduced to study the maximum of the corresponding CTMMP {(Jt, St)}t>o in 47 terms of Asmussen's methods, which will be introduced in the following. First, the existence of the maximal aggregate loss distribution for the risk model we consider is contingent upon there being a finite maximum M for the perturbed CTMMP. Asmussen points out that the condition for this is y~] ^n < o, i M is contingent upon an average negative drift of the CTMMP {(Jt, St)}t>o, which is equivalent to a positive security loading in the associated risk process. This condition was assumed in the models described in the previous section. Figure 3.1: Coupling of aggregate loss and fluid flow processes. For all non-negative levels x < M, Asmussen defines m(x) to be the state of the underlying Markov chain at the first instance that the process reaches level x 48 (see Figure 3.1). He then establishes the following: Proposition 3.3.1 {m(x)} is a terminating Markov jump process with life-length coinciding with M. The consequence for the maximal aggregate loss for our perturbed risk process is that it can be viewed as a terminating Markov process, and as such, M follows a phase-type distribution for the full generality of our model. Therefore, 4J{U) = F(M >u) = aeVue, (3.7) where a is the probability vector same as that of J0. In what follows, we identify the intensity matrix U for that distribution. 3.3.1 Determining U in the Pure Brownian Case We suppose initially that the process is subject to Brownian motion in all states, i.e., of > 0 for each underlying state i. We know that the process {St}t>o is modulated by the continuous time Markov chain {Jt}t>o, and to analyze it, we need to investigate the behavior of it during each residence time in the successive underlying states, which is an exponential time period. That is just the distributional impact of the evolution of the Brownian motion during exponentially-distributed intervals, which in turn can be assessed due to the following result (for example, see Bertoin [19]). 2 Lemma 3.3.2 Let {£t} be Brownian motion with variance a and drift r, and let T be an exponential random variable with rate [i which is independent of {&}• Then the random variables 49 Y = max \ft — £ri; X = max {&} 0 r r 2u , r r 2u . r? = - + v- + -? and uj = ~- + \— + -?> 3-8 respectively. In other words, the amount of the maximum Y of the Brownian motion over the exponential time T and the amount of drop Y from that maximum to its final value are independent, exponentially distributed random variables, with respective rates u and r\. In applying this result to a sojourn in state i, after substituting for /J, = Xi,a2 = of and r = r* in (3.8), one obtains r2 2A- Asmussen [10] then exploits the independence of these two random variables, and works with the underlying structure of {(Jt, St)}t>o as a succession of visits to the various states i, each of exponential duration, to establish that my u y Uij = -Ui 5tj + Ui I rji e~ d.y V] pik e^e ej , (3.10) JO keE,.,-n , tb where pik is the (i, k) entry of the matrix P = I + A1/MA 50 After substitutions are made, the foregoing result says that the ith row of matrix U th _1 equals the i row of matrix A2\/a2P(r]iI—U) —Au. which is the first result of his Theorem 4.1. A more tractable formula for U by making use of the uniformization technique is presented in his Theorem 4.2 that works with a single rj > maxjjr^j}. Precisely, let r\ be large enough such that and that fM > ~[A]« Vi. (3.12) Then U satisfies the non-linear matrix equation U = ipo(XJ), where 1 ^o(V) = {A2fl/ U = lim U(n) where U(0) = - Au and U(n + 1) = Vo(U(ra)). n—>oo For one application of this result, we will apply this algorithm to find the mean busy period of the perturbed fluid flow model in Chapter 5. 3.3.2 Determining U in the Partially Perturbed Case Our case, however, corresponds to Asmussen's "mixed" the Brownian motion does not apply during periods in the states of E+ (which reflect the payment 51 of claims). In our model, because we take the premium rate as the drift of the Brownian motion, we don't have linearly decreasing phases. Therefore, the infinitesimal generator U of M can be partitioned to conform with A, having the block form u(<7ff) u( Further, the initial probability vector a of M is just that of m(0) since the starting phase is always from decreasing phase set Ea. Then, for i € E+ and j 6 E, the pertinent formulas are U(+*) = A(+<7) (3-14) and U(++) = A(++) (3.15) (see Asmussen [10], formula (6.1)). A^++^ refers to transitions among the un perturbed states during a single artificial time segment, while A^+a' refers to the instantaneous rate of transiting from unperturbed to perturbed states (i.e, upon completion of paying a claim). For i £ Ea, the current environment is perturbed. Hence for i 6 Ea', and j 6 E, we can refer to the results obtain in the previous section: /•oo Uij = —u>i5ij + uji ^° keE" where rj is given in (3.11) and (3.12). In matrix notation, Asmussen [10], formula 52 (6.7) shows that this implies W ff+ x Vx U ,U( >] = -(AW,0) + (A2/VCT2,0) / e-" Pe dx 1 = (AWCT2,0)P(T?I-U)- -(AW, 0) (3.16) and according to Theorem 6.1 therein, U can be computed iteratively as follows. Define matrix function / y(^) \(<7+) \ ^i(V) = <+<*) T(++) V ) where V has the same dimension as U, and Y^a<7' and Y^a+> are determined by Equation (3.16), i.e., Vx v(ffff))V(ff+)\ = _(Awj0) + (A2p/(T2,0) / e-^Pe dx 1 (A2WCT2,0)P(r?I-V)- -(Aaj, 0). Then U = lim U(n) n—>oo where U(n + l) = ^i(U(n)) with '-^ o N U(0) T(+(J) T(++) 53 Having obtained U, the ruin probability is given by i)(u) =P(M >u) = aeVue The probabilities of ruin due to diffusion and claims, respectively, are given by I ACT) \ I o ^ Vu 4> {u) = a eu « ipd(u) = ae c e<+) \ ° ) v ; The former is due to the fact that the ruin occurs by diffusion only when the phase of the underlying Markov process {Jt}t>o is in phase set Ea\ whereas for the latter case caused by a claim, {Jt}t>o is in phase set E+. 3.4 Numerical Examples Example 3.4.1 The first example we present is from Li & Garrido [41], who considered a Sparre-Andersen risk process perturbed by an independent diffusion process, in which inter-arrival times have a generalized Erlang(2) distribution with rate parameters Ai = 1.5 and A2 = 3 and claim sizes are exponential with rate (3=1. The premium c = 1.1 and the infinitesimal variance a2 = 1. We have compared our results for a number of values of u, and the results are consistent in all cases studied (Table 3.1). Example 3.4.2 The next example we considered is the "contagion" example de scribed in Section 3.2.3. a^ = 0.02 and as = 10. Other parameters are listed in Table 3.2. In Table 3.3, we compare the ultimate ruin probability, denoted as i/j1^IAPd, 54 Table 3.1: Comparison of ruin probabilities u 4>s{u) ipd(u) ijj(u) L & G Ours L & G Ours LkG Ours 20 0.1390 0.1388 0.07171 0.07171 0.2107 0.2105 40 0.03091 0.03086 0.01595 0.01595 0.04686 0.04681 60 0.006875 0.006864 0.003547 0.003547 0.01042 0.01041 80 0.001529 0.001527 0.0007890 0.0007888 0.002318 0.002315 Table 3.2: Parameters for contagion example Env 1 Env 2 Inter-arrival time Ai = l Ai + A2 = ll Ai c-m x j A2 c~U9.x claim size (pdf) A1+A2 A1+A2 Mi = 5 ^2 = 5 premium (c) 1 1 for the perturbed model with a very small variance with the corresponding non- perturbed values from Badescu et al. [16] (probability of ruin before time 100, 000). For initial surplus levels u = 1 and u = 10, a high degree of agreement is observed. Figure 3.2 shows the ruin probabilities as functions of the infinitesimal standard deviation a. The results are quite intuitive: an increase in the variability of the drift component increases the overall ruin probability. However, the portion of the ruin probability due to claims actually decreases. An intuitive explanation is that some sample paths that lead to ruin in the unperturbed model would encounter ruin earlier in the presence of perturbation. Example 3.4.3 The third example is taken from Lu & Tsai [44]. It studies the ruin probabilities in a Markov-modulated environment. The environment process is a two-state Markov process with transition rates a\ = 1/3 and a Table 3.3: Comparison of ruin probabilities (a = 10 u == 1 u = 10 Mt)MAP ^MAPd Mt)MAP ^MAPd t = 100000 0.846654 0.784840 t — oo 0.846654 0.784839 ^a(lO) 0.8 0.6 0.4 0.2 Figure 3.2: Decomposition of ruin probabilities varying according to a. Solid: total ruin probability; Green solid: due to claim; Dashed: due to diffusion information about the inter-arrival time distribution, the claim size distribution, the premium rates and the infinitesimal variances are listed in Table 3.4. In this case, the phase generator for m(x) is given by -0.801367 0.0906816 0.23294 0.223728 0.0508765 0.10733 1.01093 -3.42545 0.136478 0.210178 1.19299 0.741361 0 0 -1 1 0 0 U 1 0 0 -1 0 0 0 2 0 0 -2 0 0 0.5 0 0 0 -0.5 56 Table 3.4: Parameters used in the model State 1 K = 1/3) 2 (a2 = 2/3) Claim rate 0.5 1 Claim size (pdf) p2xe~f3x P = l £ = 0.8, ft = 2, fo = \ Premium (c) 1.35 1.35 Infinitesimal variance 4 1 |(r(u) Ik 2.5 5 7^5 10 12.5 15 Figure 3.3: Decomposition of the ruin probability. Light solid: by our method; Dashed: from Lu & Tsai [44]. With the generator U, the ruin probabilities can be calculated from equation (3.3.2). Figure 3.3 shows the total probabilities of ruin and the probabilities of ruin due to diffusion and claims when the environment process starts in state 2. Results using our method and Lu & Tsai's method are both plotted. As the figure illustrates, the two method produce identical results. One advantage of the method proposed in this paper is that same routine may be applied straightforwardly to more complicated situations. 57 Chapter 4 ERLANGIAN APPROXIMATION TO FINITE TIME RUIN PROBABILITIES IN PERTURBED RISK MODELS 4.1 Introduction As we did in Chapter 3, in this chapter, we consider the aggregate loss model1 Nt St = Y,Zk-ct-aiu (4.1) fc=l where c > 0 is the premium received per time unit, {Zi\ i = 1, 2,...} are indepen dent non-negative claim random variables, the process {Nt}t>o counts the number of claims up to time t, a > 0, and {£t}t>o is a standard Brownian motion. Further, we assume that the claim amounts Zf, i = 1, 2,... are phase-type distributed. In addition, in the development which follows, we will let the other parameters of the model (the inter-claim times, the premium rate, and the variability parameter a) depend upon the state of an underlying Markov chain. 1The material in this chapter has been accepted for publication in Scandinavian Actuarial Journal. 58 The probability of ruin before time T is given by ij(u,T)=P(r(u) We know from Chapter 3 that the ultimate ruin probability I/J(U) given in (3.2) for this model has been studied extensively in the literature. See for example, Dufrense & Gerber [27] and Tsai & Willmot [63] for the case where {Nt}t>o is a Poisson process, Li & Garrido [41] for the case where {Nt}t>0 is a renewal process with interclaim times following a generalized Erlang distribution, and Lu [44] for the case where {Nt}t>o is a Markov modulated Poisson process. The methods used in the foregoing papers are all based on constructing and solving defective renewal equations, yielding formulas for ip(u) or related quantities. Usually, the exact determination of the finite time ruin probability is considered much harder than that of its infinite-time counterpart. Efforts have been made to obtain ip{u,T) in various non-perturbed risk models for some time, along a variety of lines. Thorin & Wikstad ([61] and [62]) used a root finding method for selected Sparre-Andersen models. Exact solutions for ip(u,T) in the classical risk model based on evaluating Bessel functions were obtained by Drekic &; Willmot [25] in the case of exponentially-distributed claim sizes. By discretising time, Dickson & Waters [23] obtained a recursive formula at successive time instants for xjj(u,T) for classical models. Stanford & Stroihski [57] and Stanford et al. [58] attempted to avoid discretising time for ip(u, T) in both classical models and a few Sparre-Andersen models. Instead, they embedded the recursion at claim epochs. Asmussen, Avram & Usabel [13] developed the so-called "Erlangization" technique to approximate the finite time ruin probability for the classical risk model with 59 phase type claim sizes, while Stanford et al. [55] applied Erlangization to the Sparre-Andersen and stationary renewal risk model cases. The main purpose of this chapter is to provide an efficient procedure to approx imate the finite time ruin probability ip(u,T) for a fairly large class of perturbed risk models by making use of the Erlangization technique, and the approximation is asymptotically exact as the order of the Erlang increases. We believe the present work, alongside a Martingale approach of Badescu & Breuer [15], represent the first attempts to either compute or approximate finite time ruin probabilities in the perturbed risk model. The main reason why it is typically so much harder to determine ip(u,T) than ip(u) is that one loses the renewal structure that an infinite timeline provides. Er langization circumvents this problem by replacing fixed time intervals with random, open-ended intervals with the same means as the finite time periods of interest. In so doing, the renewal structure is regained. The simplest such random interval is of course an exponential distribution with mean T. A simple generalization of it is the family of Erlang distributions comprising L successive exponential stages, each of which has rate parameter equal to L times the original, so that the overall mean is still T. The sequence of Erlang distributions Er(L), L = 1,2,..., with a common mean T converges to a point mass at T of probability one as L -—> oo (Figure 4.1). (Also see, for instance, Kleinrock [38], pp 124-125.) In order to exploit the solution method we propose below, it is necessary to associate the risk model {St}t>0 defined in (4.1) with a sample-path-equivalent continuous-time Markov-modulated process (CTMMP). This CTMMP has rates of increase and decrease based on the underlying state of the associated continuous- time Markov chain {Jt}t>o- For risk models that can be accommodated in this way, 60 The family of L-phase Erlangian distribution Er(L) Figure 4.1: The family of L-phase Erlangian distributions with common mean T. we construct a composite process that keeps track of both the states of {Jt}t>o as well as the Erlang stages. Then by exploiting established results for CTMMPs, we are able to determine our accurate approximation to ip{u,T). Last but not least, we develop an efficient procedure to obtain the approximation of ip(u,T) by recognizing a repeating structure in the probability matrices we work with. Each of these points is described in the sections which follow. We observe that the method presented herein to approximate ip(u,T) is appli cable to any risk model which can be associated with a sample-path-equivalent CTMMP. The remainder of the chapter is organized as follows. In the next section, we explain Erlangization and the construction of our composite process. Section 4.3 states the resulting form of the ruin probability and its components due to drift and claims. Section 4.4 presents a much more computationally efficient algorithm, 61 by exploiting the repetitive block structure of the matrices involved. Numerical examples are presented in Section 4.5. 4.2 Erlangization &; Construction of the Com posite Process In Chapter 3, we presented a method to obtain the ruin probability for perturbed risk processes in terms of their sample-path-equivalent CTMMPs. In what follows, we consider the finite time ruin probability for the same class of risk models by making use of Erlangization. This requires us to construct a composite process for the CTMMP that comprises the risk process and an independent Erlangian distribution, the details of which are explained in what follows. The central idea behind Erlangization is to approximate the fixed time inter val of length T by a random interval whose mean is T. By letting this interval comprise L of i.i.d. exponential stages, a sequence of approximations obtained which converges almost surely to the fixed time interval as L —> oo. Thus, the probability of the time of ruin before time T can be approximated by any number of the sequence of ruin probabilities prior to the Erlang horizon. This sequence of Erlang distributions is chosen to have common mean T and increasing number of stages (see, for instance, Theorem 6 of Asmussen et al. [13]). (It needs to be noted, however, that the rate of convergence gets progressively slower.) To this end, we first use a general phase-type distribution PH(/3, H) as the time horizon and construct a composite process from process {(Jt, S)}t>o and PH(/3, H). More preciously, let {Jf}t>o denote the underlying Markov chain for the phase- type distribution PH(/3, H). Assume that it has state space En with order p#. The 62 composite process, denoted as {{Jf, Jt)}t>o is constructed by pairing each of the phases of the "horizon" PH(/3, H) with its concurrent phase i for {Jt}t>o at time t. This Markov process has state space E = E° U EH" U EH+ with E° = {C} for the absorbing state once the horizon has been reached (C here denoting "cemetary"), EH" = EH x Ea and EH+ = EH x E+. Working with this construction, the probability of an event of interest occurring by time T can be approximated by the probability of the event occurring while the composite process resides among the transient states. H The evolution of the process {St}t>o is as follows. When jf- = i G E , Jt — j € 7 E" , {St}t>o behaves like a Brownian motion with negative drift Cj and infinitesimal + variance cr?. In contrast, when Jt = j e E and no matter the value of J^, it merely increases at the unit rate. So the vector of the drifts and infinitesimal are variances of process {St}t>o in all possible states of {{Jf, Jt)}t>o given by rH = eH <8> r (4.2) and 2 cr% = eH (8) cr , (4.3) where ® represents the Kronecker product (see Graham [34]). It is obvious that the process {(o has initial distribution (0 f3 <8> a). a While Jt G E , it is possible for {J^}t>o to change state. In contrast, when Jt G E+, there is no passage of real time and consequently, the current value of {J^}t>o among the transient stages must be preserved. Therefore, the generator 63 ff of the process {(Jt , Jt)}t>o is given by 0 0 0 } +) ( + ( /^H'") A(ff"+) ^ *-H *-H k-tf (4.4) V A In the special case where PH(/3, H) is of order 1, the time horizon is exponential with rate a (say), we have / A^-al A^ Aff = (4.5) V A(+-) A(++) I 4.3 The Ruin Probability Prior to a Phase-Type Time Horizon Similar to Section 3.3, we define the record high phase process {mH{x)}x>o to be the state of the underlying phase process {(Jf, Jt)}t>o at the first instance that the process {St}t>0 is at the maximal value x. Then {mH{x)}x>o is a terminating 64 Markov process whose generator can be partitioned into / Tf(H<">) TT(H"+) \ u H u H U H (4.6) v <•' <**> j Once XJH has been obtained, based on Theorem 1 in Asmussen et al. [13], the probability of the time of ruin before time horizon PH(/3, H) is given by V>(u,H) = (/3®a)eu'u"e. (4.7) As in Dufrense and Gerber [27], we can decompose the ruin probability by its cause. The probability of ruin due to diffusion is given by / .,„(<0, \ ip(u,H) = ((3®a)eu-VH \ Q(+)^ J while the probability of ruin due to claims is given by l QW » ^(u,H) = {j3 The former is due to the fact that the ruin occurs by diffusion only when the phase of the underlying Markov process {Jt}t>o is in the decreasing phase set, Ea; whereas for the latter case caused by a claim , {Jt}t>o is in ascending phase set, E+. The approach for determining the matrix U# follows the same steps described in Section 3.3.1, now applied to the composite process. Specifically, by substituting 65 parameters r# and er# into formula (3.11), and ensuring that the resulting TJH satisfies (3.12) for rate matrix A# given in (4.4), we obtain the vectors /% and UJH which are used for computing U#. It is easily seen that /J,H and u># have the same structures as those of r# and er#, which we choose to express as HH = eH ® fi', U>H = eH ® u>' • where e# is a column vector of ones, while /z' and u>' are appropriately dimensioned column vectors containing the individual elements /^, u[. Then, the direct application of the algorithm presented in Section 3.3.1 to the matrix U# yields the following nonlinear matrix equation: ( vW*) vV*a+) \ vH = MUH) •= v£ ' n (U (H+°) ; A (U++) \A>HAW " A H where matrices YH and YH are determined by the following equation A„ 0^+>), (4.9) and where Q +) + Q(f = A1//,H.A^ )=IH®(A1/M,.A^)). 66 In addition, U# can be computed by the following iteration scheme: VH(n + l) = fa(UH(n)), (4.10) where the initial value U#(0) is given by Uff(0) (H+°) J A (H++) \ A£AH " A H 4.4 Simplifications for Erlang Horizons One challenge in using the recursive formulas in the last section is that the size of the matrix U# grows quickly as the Erlang horizon matrix H does. However, with an Erlang horizon, this problem can be overcome, since the process can only move forward one stage at a time, and never backwards. This yields a Ug that is block upper triangular in form, with a special structure. Asmussen et al. [13] show that when Erlangization is applied to the classical risk model to approximate the finite time ruin probability, the "upcrossing phase probabilities" matrix has a so-called "block Toeplitz structure" and derive a recursive formula to calculate the entries of it. We define "block Toeplitz structure" as a block upper-triangular matrix such that Vi < j, the (i,j)th block is the same for all blocks with the same index j — i. Thus the main diagonal blocks, super-diagonal blocks, and so on, each feature a single, repeating block. Our matrix U# possesses this structure as well, and the latter part of this section presents the equations to obtain the successive blocks. As a result, rather than calculating one very large matrix, it suffices to compute L smaller block 67 elements of U# that repeat, and therefore the computational expense is greatly reduced. Recently, Ramaswami et al. [49] extend these results to a general fluid flow model with Erlang horizon incorporated. In this section, we further extend these results to the partially perturbed fluid flow case. 4.4.1 The Structure of the Matrix UH In this section, we present the special structure of the matrix U#. The corre sponding result for the classical risk model first appeared in [13]. Theorem 4.4.1 When the time horizon has Erlang distribution Er(L), the block matrices LrH and LrH in UH have L x L-block Toeplitz structures. That is, ( \ UQ UX ... uL_2 uL_x a) TA™) fA.™) 0 Xjt uL-3 uL-2 0 0 a) 0 0 o u£ and ( ut+) v 0 0 CT+) 0 0 0 t/0 j where +) ^ and lff are matrices with dimensions \Ea\ x \Ea\ and \Ea\ x \E+\ respectively. To prove the result, we first state a lemma related to the block Toeplitz struc ture matrices, which can be easily verified by algebra calculations. Lemma 4.4.2 Assume that matrices A, B and X have L x L-level block Toeplitz structures. 1. If matrix product A0B0 is conformable, then AB has L x L-level Toeplitz structure as well. Further, the (£, m)th block entry, {i < m, £, m = 1,2,..., L), of AB is AoBm-£ + AiBm-e-i + ... + Am^Bo m-t = / j •"•i-t'm—i—i • i=0 2. If X has inverse Y, then Y is of L x L-level block Toeplitz structure and YQ = X0 , Yt = -(YjYiXt-Axt1, (t = l,2,...,L-l). Proof of Theorem 4.4.1. Since the initial value U#(0) is given by ••UlH Uff(0) = V AST' A Toeplitz structure, we just need to show that if U# (n) and XJ£ (n) have L x L- 69 level block Toeplitz structures, then U# (n+1) and U# (n +1) have L x L-level block Toeplitz structures as well. From (4.9), we have for n > 0 Ufln+l) Uf+>(n + l); CT+ IH ® (A2/V/(T2 + A2/(T2A^) + H «g) A2/o.2, lH 7?ffI - Uff (n)) ~* - (lff ® Aw, IH ® 0^) . (4.11) Three different types of matrix terms can be identified from the right hand 1 side of (4.11): (T]H1 — Un(n))^ , I# ® Z and H ® A2/o-2, where Z is some block matrix. Since the matrix H and I#- trivially have Toeplitz structures, so do the terms I# ® Z and H <8> A2/(T2. Turning to the term (r/#I — Ujy(n))-1, we first partition (77^1 — U#-(n)) into the same form as in (4.6) for U#. That is, let ',»\ ml - Uff(n) (4.12) VC D, Then, matrices A, B, C and D all have L x L-level block Toeplitz structures and each has block submatrices with dimensions \E"\ x \Ea\, \Ea\ x |.£7+|, \E+\ x \Ea\ and |.E+| x \E+\ respectively. Define the Schur complement SA of A by S^ = D — CA_1B (see, for example, Abadir & Magnus [1]). The Schur complement enables us to state the inverse of 70 ?7tfI-Utf(n) as \ / A B 7]Hl - Uff(n) yc D; A"1 + A^BS^CA-1 -A_1BS- -^ (4.13) -S^CA-1 c-i V 3A / Thus, each of the block matrices has a Toeplitz structure because of the Lemma's provisions. Combining all the terms and using the Lemma, we conclude that the block matrices \JH and U^ in U# have Lx L block Toeplitz structures. • In the next subsection, we derive recursive formulas for calculating U^ and U^ based on the previous values U^^ and U^x , i = l,2,...,L — 1. It turns out that the matrix computation for Uj and U, only involves matrices with dimensions |£CT| x \Ea\, \Ea\ x \E+\, \E+\ x \Ea\ or \E+\ x \E+\. In contrast, in each iteration given by (4.9), the dimension of the matrices involved is (\EaUE+\ • L) x (\Ea U E+\ • L), which grows as the horizon stage increases. (HC"T) (Ha+) y 4.4.2 Recursive Formulas for XJ H ' and U# Before deriving the recursive formulas, we first state a property of the vectorization of matrices. In particular, the operator vec transforms a matrix into a vector by stacking its columns one underneath the other to form a column vector. Specifi cally, letting A be an m x n matrix and a; its ith column, then vec( A) is the mn x 1 71 vector / \ ai a2 vec(A) = Va«/ The following well known result of the vec operator will be relevant (see for example, Graham [34]). For a conformable matrix product MXN, vec(MXN) = (NT We are now in a position to derive recursive formulas for the successive blocks within XJH and XJH in U#. According to (4.8), we know that the matrix XJH is the fixed point of the quadratic matrix function ^(O- Therefore, according to Equation (4.9), 1 ur> ur+)) = Awa3H.(Qcr> qcr^i-u*)- -(A^ 0^+>). (4.15) Right multiplying on both sides of the above equation by TJHI — U# yields m^{r] u^)u(H-)+u^+)A(?++)^ = (X' Y') + (wujf^. WU^+) 72 where (OTT) = Iff ® (A2AiV + + Y' = A2/^-A^ > = Iff®(A2/ Then, by equating components in the foregoing equations, we have: ] +) CT) m\jT - (upur+u^ Ar )=*+wu + + + + mV(»° ) _ (u(f"")n^ ) + Uf )AP) = Y' + WU^ ). (4.17) T(#™ ); It sufficeluin^css tuou determinLicLciiiinie alanl oULf thune elements inn thtuec firsmot t blocuiuuikv rowmwas oufi U«_ ^ and XJH due to the repetitive structure of these matrices. For m = 0,1,..., L — 1, equating the (1, m + l)th block entry on both sides of (4.16), we find that m J2 (W, - USW>)UH - U^A^ = X^ , (4.18) i=0 th th where W^ and X'm are the (1, i + l) and (1, m + l) block entries of matrices W = ?7ijl — W and X respectively. Similarly for (4.17), we have +) + £ (w; - U^jU^ - U£ A( +> = Ym , (4.19) i=0 73 where Y^ is the (1, m + l)th block entry of Y', m = 1,2,... ,L — 1. To solve for the blocks U^ and U^+), we rewrite equations (4.18) and (4.19) as +) (+CT) w'0 - uft^ju^i^ - i^u^u^ - I(-)U^ A m— 1 x^ - £ w'v{z\ + E uiw)uS3 i=l i=l and <7ff) ++ ++ (W0 - U0 )u^+)l( ) - I^U^U^ - I(-)U^A( ) m—1 (crcr)TT(cr+) = n-^wiu^ + ^uj^um—. i ' i=l i=l We have retained the matrices V-™' in the matrix triplets above, in order to exploit (4.14). Taking vec operations on both sides of the foregoing two equations and using formula (4.14), we have, for m = 1, 2,..., L — 1, 1^ (+*) I^) A vec U^n J m—1 (o-cr)TT(o-ff) vec X^-^WjuB + ^U^U m—i (4.20) i=l 74 and ,T J( m—1 vec Y^ - X; WJUK + £ Uf^ua (4.21) i=l vec(U^+' j and vecf U^0"-1 J are found by solving equations (4.20) and (4.21), and from these we need to merely reconstruct the matrices U^+) and U^. More precisely, let A! i^ = (A(+CT)) ®I(<7 771—1 vec w CTCT)((T(7)TT(<7CT) = K - E ^ + E U! U m—i i=\ denote the three matrices in (4.20), ++) } ++ T A2 i< ® (w(, - u[r ) - (A( )) ® i^\ (CT+A ^T(^) B2 = u, m—1 w) (-+) Dm vec Y'm - X; W5U<2 + X ui U. i=i i=l denote the three matrices in (4.21). We have the following result: Theorem 4.4.3 For m = 1, 2,..., L - 1, + wee £/£ >) = f A2 - BzA^BiV (A„ + B^C, (4.22) 75 and a) +) vec lt ) = V (Bi vec( lt ) + Cm) . (4.23) Notice that the matrices Cm and Dm involve Uj and U] for i = 1, • • • , m- 1. The following matrices necessary for the recursion are initialized as follows: W0 = Jjffl-A^, W'm = 0, for m = 1,2, ...,L-1, x'o = ^+V'AW-AV-^A" X^ = A A2/ff2, X^ = 0, for m = 2, 3,..., L — 1, + Y'0 = A2/^-A^ \ Y' = 0, for m = 1, 2,.... L - 1. The initial values U0 and UQ correspond to the generator of the record height 1 process {m#(x)}x>o for the composite process {(J^ , Jt),St}t>o with exponential horizon of rate L/T. Thus, we may apply the procedures described in Section 4.3 to this particular CTMMP with generator given by (4.5) with a = L/T to obtain U^andU^. Remark 4.4.1 As a quick comparison of the running time for the two methods (one being the recursive fashion given in Theorem 4-4-3, and the other being the one described in Section 4-3), we computed the UH matrix using Mathematica for Example 4-5-2 below. The running times for these two methods are listed in Table 4-1 for L = 3, L = 9 and L = 20 (rounded to the closest thousandth of a second). 76 The drastic reduction in run time using the recursive method is apparent. The Mathematica notebooks were running on an IBM T30 laptop. Table 4.1: Comparison of the running time (T = 1000, u = 100) a L = 3 L = 9 L = 20 non-recur. recur. non-recur. recur. non-recur recur. {4.0,4.0} 6.029 < 0.01 30.745 <0.01 121.348 0.03 {1.0,1.0} 8.783 < 0.01 40.879 0.01 158.808 0.03 {0.01,0.01} 15.613 0.01 73.106 0.01 277.760 0.03 4.5 Examples We use two examples to illustrate the methodology described in this chapter. Example 4.5.1 This example repeats the "Contagion" example in Badescu et al [16]. In this example, two environmental conditions control the claim rates and severities. Environment A corresponds to a normal situation with standard claim rates and claim sizes, while environment B reflects periods of contagion, when a highly infectious disease is causing a supplemental stream of claims. The environment switches from A to B at rate a A and from B to A at rate as- We consider two sets of infinitesimal variances: a — {10,1} and a = {0.01,0.01}, while the latter one can be thought of the approximation to non-diffusion case. In addition, as in Asmussen et. al [13], we use the Richardson extrapolation to improve the estimate 77 where T is the fixed time, L + k and L are stages of Erlang horizons. Using the parameter values given in Badescu et al. [16], we calculated the probability of ruin before T = 100. We then compared our results to those obtained in Badescu et al [16] for the non-diffusion case, and Badescu & Breuer [15] for the diffusion case (both of which are obtained via Laplace transform methods). Table 4.2: Finite ruin time probabilities by Erlangian Approximation (t = 100, u= 10) L a = {0.01,0.01} Extrapolation a = {10,1} Extrapolation 1 0.57230 — 0.89870 — 3 0.60935 0.62782 0.92538 0.93873 5 0.61766 0.63014 0.92877 0.93386 7 0.62130 0.63040 0.93009 0.93338 10 0.62405 0.63046 0.93103 0.93321 15 0.62619 0.63047 0.93173 0.93313 20 0.62726 0.63047 0.93207 0.93309 25 0.62790 0.63047 0.93227 0.93308 30 0.62833 0.63047 0.93241 0.93308 Badescu et al. [16] 0.63034 Badescu et al. [15] 0.93306 The results are shown in Table 4.2. In both cases, the approximation is already quite reasonable for even small values of L. From the table, we can see that with a = {0.01, 0.01}, the probabilities of ruin are slightly bigger than those obtained by Badescu et al. [16] for the non diffusion case. This is because our small diffusion component gives rise to more randomness in the model. However, the approximation in the non-diffusion case is even better, and the results improve further by making use of the Richardson extrapolation (based on the given row and the preceding row in the table). Figure 4.2 shows how the total ruin probability, as well as the probabilities of 78 (Mio) 0.8 0.6 0.4 0.2 _^!!I_ , , , . , , , , , , , _ ^ g 2 4 6 8 Figure 4.2: Decomposition of ruin probabilities. Solid: total ruin probability; Green solid: probability of ruin due to claims; Dashed: probability of ruin due to diffusion ruin due to claims and due to diffusion changes with the infinitesimal variances. The parameters we used are u — 10 and L = 2. Obviously, as a gets bigger, both the total probability of ruin and the probability of ruin due to diffusion increase. Example 4.5.2 In this example, as in Stanford et al. [55], we consider the Sparre- Andersen example presented in Thorin & Wikstad [61], Table 8. We calculated the ruin probability with different values of the infinitesimal variances. The results are listed in Table 4.3. It shows that the ruin probabilities increases as the infinitesimal variance increases. When the infinitesimal variance is 0.01, the ruin probability is quite close to those obtained by Stanford et al. (Table 1 in [55]), which is shown in the last row of the table. 79 Table 4.3: Finite ruin probabilities by Erlangian Approximation (T 1000, u 100) a L = 1 L = 3 L = 5 L = 7 L = 9 4.0 0.36834 0.42794 0.44227 0.44855 0.45205 3.0 0.31575 0.36592 0.37849 0.38412 0.38729 2.5 0.29293 0.33849 0.35006 0.35527 0.35823 2.0 0.27374 0.31524 0.32584 0.33065 0.33339 1.5 0.25874 0.29700 0.30679 0.31125 0.31380 1.0 0.24814 0.28410 0.29330 0.29749 0.29989 0.5 0.24184 0.27645 0.28529 0.28932 0.29163 0.2 0.24008 0.27431 0.28305 0.28704 0.28932 0.1 0.23983 0.27400 0.28273 0.28672 0.28899 0.01 0.23975 0.27390 0.28263 0.28661 0.28888 Stanford et al. [55] 0.23975 0.27390 0.28263 0.28661 0.28888 80 Chapter 5 THE MOMENTS OF THE RUIN TIME IN MARKOVIAN RISK MODELS 5.1 Introduction In this chapter, we consider the question of determining moments of first passage times for two models: One is for the time of ruin for an aggregate loss risk process and the other for the mean busy period of the perturbed fluid flows. Besides the importance of the question itself, the other purpose for studying these quantities in these two respective models is that we want to present a step in the direction of obtaining moments of the time of ruin in perturbed risk processes. The theoretical foundations and the computational procedures are similar for these two models. In the rest of this section, we introduce these two models in detail. Aggregate loss risk processes Consider the following general aggregate loss risk process in continuous time, 81 denoted by {St}t>o, with St = Y^Zk-ct, t>0, (5.1) fc=i where c > 0 is the premium received per time unit, {Z^i = 1,2,...} are inde pendent non-negative claim random variables and the process {Nt}t>o counts the number of claims up to time t. The time of ruin is T(U) = inf{£ > 0 : St > u}, where u is the initial surplus of the insurance company, with r(u) = oo if ruin never occurs. Therefore, define kth moments of the time of ruin to be k fi^ = E[r(u) • I(T(U) < OO)], k = 0,1,2,... , where I(s/) denotes the indicator function of the event stf, which is equal to 1 if the event occurs, and 0 otherwise. There are several results for the moments of the time of ruin in the literature. In the classical risk model setting, Lin k, Willmot [43] obtain the kth moment of the time of ruin in a recursive fashion with claim sizes being general i.i.d. random variables. The starting point in their method is a general renewal equation (see Lin h Willmot [42]). Drekic & Willmot [25] further give a simple expression for the kth moment of the time of ruin under the assumption that the claim sizes are i.i.d. exponentials. They invert the LST of the time of ruin and directly obtain the distribution function of the time of ruin. Drekic & Willmot [26] then extend 82 the result to the situation where the claim sizes are phase-type distributed. In paper [24], Drekic, Stafford k, Willmot also use symbolic computation to combat the difficulties appearing in the computation of the moments by making use of the formula given in [43]. For the non-classical risk model setting, Dickson k. Hipp [22] consider an Er- lang(2) risk process as inter-claim time process. Recently, Albrecher &; Boxma [8] study the Gerber-Shiu discounted penalty function in a semi-Markovian risk model (for definition and discussion of the Gerber-Shiu function, see [31] and [32]). In both papers, authors first derive LST of the time of ruin, then obtain moments of the time of ruin by taking derivatives of those LSTs. To get the kth moment, one needs to apply root finding methods. In the case of the Albrecher & Boxma formula, a root finding method needs to be used in each recursive step. From a computational point of view, it is not convenient. In this chapter, we come up with an approach based on matrix-analytic meth ods to obtain moments of the time of ruin defined by (5.1), which is applicable to general Markovian risk models. The class includes the Sparre-Andersen risk models with both inter-claim time and claim sizes being phase-type distributed, and Markov-modulated risk models with claim sizes being phase-type distributed. Same as the methods presented in previous chapters, the proposed approach is readily programmable and computationally stable. The only limitation for the risk process under our approach is that it must have a sample-path-equivalent CTMMP counterpart. These details will be further clarified in Section 5.2. We conclude this section with focus on perturbed Markov-modulated fluid flow mod els. Perturbed fluid flow models 83 The perturbed Markov-modulated fluid flow process comprises two components. One is a flow level process {Vt}t>o which evolves according to Brownian motions during specific time periods, and the other is a finite continuous time Markov chain {Jt}t>o (also called the environmental phase process) which modulates the fluid flow {Vt}t>o in such a way that, when the phase process {Jt}t>o is in state i, the level process {Vt}t>o evolves according to a Brownian motion with the linear drift 7-3 and the infinitesimal variance of (cr, > 0). A particular sample path of {V^}t>o is given in Figure 5.1. In Figure 5.1, {Jt}t>o sojourns in three environmental phases as time goes by. Three time periods are illustrated as circle, solid and dashed lines, respectively, and the linear trends in first and third time periods are upwards and downwards in the middle one. Figure 5.1: \E\ = 3. The three environmental states are denoted circle line, solid line and dashed line A fundamental quantity of interest in the literature of stochastic fluid flow models is the busy period of the flow, which is defined as P(u) = inf{t > 0 : Vt = 0 | V0 = it}. (5.2) 84 where u is the initial flow level. We note that, due to diffusion, the above definition of the busy period becomes trivial when the initial level u = 0, so that we have to start the busy period at level u > 0. The stability condition for the level process {Vt}t>o is that the average drift is negative, that is, ^ KiTi < 0, where E is the state space of {Jt}t>o and TV = {7rj}ieB is the stationary distribution of {Jt}t>o- This stability condition is assumed throughout the chapter. The special case of at = 0 has been heavily considered in the literature by many authors for modeling data communication channels, and the main task under those considerations focus on the study of the stationary distribution of the buffer content and their related computation issues ([10], [21], [47]), and the recent contributions made by Ahn and Ramaswami et al. can be found in [4], [5], [6] and [7], and references therein. One of the interesting applications of these results to risk theory can also be found in [2], [3], [16], [17] and [48]. The perturbed fluid flow models have been paid attention as well in data com munication society, see [10], [33], [36], [37] and [50]. The study of the busy period behavior can go back to the early work by Takacs [60] for queueing systems. For Markov-modulated fluid flow settings, Asmussen [9] considers, among others, the properties of the busy period P(u). He obtains the explicit formulas for the mean of P(u) in terms of the transition matrix and linear trends of the underlying Markov chain and the Laplace-Stieltjes transform (LST) of it by making use of a change of measure technique. Recently, in a series of papers [5], [6] and [7], Ahn and Ramaswami analyze the transient behavior of the fluid flow by expressing the LST of the flow in terms of the LST of the busy 85 period. It should also be pointed out that although in [9] Asmussen states that "though not developed in detail, most of the analysis of the present paper carries over to fluid models with Brownian noise", we observe that our analysis is completely different from the method used in [9]. The key step in our study is to formulate the LST of the busy period, from which the mean is obtained by differentiating the LST. 5.2 Mathematical Preliminaries In this section, we discuss the mathematical preliminaries for the study of the moments of the time of ruin of the aggregate loss processes defined by (5.1). As stated in Introduction, Section 5.1, the methodologies developed in this chapter for the computation of the moments are quite general. Recall that the methods developed in Chapters 3 and 4 are applicable to risk processes which have sample- path-equivalent CTMMPs. Similarly, the set of risk processes we consider in this chapter is almost the same as that in Chapters 3 and 4 except that the processes are non-perturbed, i.e., a = 0. As pointed out in Chapter 2, the construction pro cedures of the sample-path-equivalent CTMMPs are the same for both perturbed and non-perturbed risk processes. Here, we briefly describe the construction steps. We start with a Markovian process {Jt}t>o which modulates the inter-claim time and the claim sizes of the aggregate loss process. The states of this process are grouped into two subsets: the set E+ of phases for the claim sizes and E~ for the inter-claim time; the infinitesimal generator of {Jt} is correspondingly partitioned 86 as / A(++) A(+-) A = (5.3) A<"+> A(" V ) We call E+ the set of ascending phases and E~ the set of descending phases. The evolution in time of the aggregate loss process {5t}t>o can be related to the evolution of a continuous-time Markov-modulated process (CTMMP) in terms of {Jt}t>o- The associated CTMMP is denoted as {(Jt, St)}t>o- The constructed CTMMP has rates of increase and decrease based on the underlying states of the associated CTMC {Jt}t>o- For risk models given by (5.1) the correspondence can be accommodated in this way, claim payments correspond to the flow increasing periods and inter-claim time to the decreasing ones. The decreasing rate of {St}t>o during the decreasing phase % € E~ of {Jt}t>o is same as the premium rate when Jt = i. Since the payment is payed at an instant in time, the increasing time periods for {(Jt, St)}t>o are "artificial" and we choose all corresponding increasing rates to be 1. Obviously, the evolution of {(Jt, St)}t>o always starts from a decreasing phase. I r(+) \ Denote r = as the vector of the linear rates of {(Jt, St)}t>o, where r(-) r(+) and r^~^ are the vectors of the rates when the phase process {Jt}t>o are in E+ 87 and E respectively, and define / A(++) A(+-) \ V A(-+) A(~) X(++) T(+_) (5.4) V T(-+) T(—) where Ai/|rj is a diagonal matrix with elements l/|r| on the diagonal. Since during the payment time period, the evolution of {St}t>o increases at unit rate, i.e., r^+^ = e^+\ we have T(+-)=A(+-) and T(++)=A(++). In the rest of the chapter, we consider risk processes given by (5.1) which have sample-path-equivalent CTMMPs. The illustrative examples of such risk processes are the Sparre-Andersen model and the Markov-modulated risk model. The construction steps of the sample- path-equivalent fluid flows were already given in Section 2.4. 5.2.1 A First Passage Time before an Exponential Horizon In order to obtain the properties of the LST of the time of ruin, we need to discuss the first passage time of {St}t>o to a positive value before an exponential horizon time H\ with rate A. The technique used here is Erlangization, which was introduced in Chapter 4. Since H\ is an Erlang horizon with order 1, according to Chapter 4, we can con- 88 struct a composite CTMMP {((Jf ,Jt),St)}t>o from {(Jt, St)}t>o and H\, where H {(Jf , Jt)}t>o is a terminating CTMC with state space EU{C} (C being the absorp tion state for {(J^1, Jt)}t>o)- From this construction, {(J^, Jt)}t>o moves exactly same as {Jt}t>o prior to random time H\, and from Section 4.2, its transition intensity matrix is 0 0 0 0 A(++) A(+~) + A A<" > A(")-AI / Then, based on Theorem 1 of Asmussen [13], we have the following proposition. Proposition 5.2.1 The probability of the time of ruin before the exponential time horizon H\ is same as the probability that the process {St}t>o will ever exceed u, where u is the initial surplus of the risk process. 5.3 Laplace—Stieljes Transform of the Time of Ruin The Laplace-Stieljes transform matrix of the time of ruin is defined by the elements sr(u) + [G(s,«)]ij =E(Uii)[e- I{T(u)<0O) jT(u)=j}\, ieE~, j £ E and s > 0, where E(U]j) denotes the conditional expectation when the initial surplus of the insurance company is u and the initial state J0 = i, and JT(„) = j is the state of the claim phase distribution when the amount of the aggregate loss first upcrosses the level u. 89 When u > 0, we decompose T(U) into two parts, one is T\{u) which is the time interval of the aggregate loss process {St}t>o being at its first upward ladder height, and the other T2(U) is the time period of {St}t>o from the end of the time interval Ti(u) to the time of ruin. It is easily seen that T\(u) is level independent, and we denote it as T\. Similar as that of T(U), for s > 0, denote \&(s) and T(s,u) as the Laplace- Stieljes transform matrices of TX and T2(U) whose elements are defined by + [*(s)]^ = Eite-^I^oo, Jri=j}], i e E~, jeE , and ST u + + \T(u, s)]zj = EM{e- ^ \T2{u)<00t j^=j}], i e E , jeE . According to Theorem 4.3 in [16], we have the following result: Theorem 5.3.1 For u > 0, G(s,u) = *(s)r(s,u), where T{s,u) = eH{s)u (5.5) with H(s) = 2<++) + T<+-^(s). (5.6) 90 Remark 5.3.1 We notice that, by the definition of^(s), for i E E~, j E E+ , [*(0)]y = Ei[e-°^I{n<00, Jri=j}] = Pj[Ti < °°> A = j], which means that W(0) is a probability matrix whose (i,j)th entry can be explained as follows: The risk process {St}t>o starts at phase i E E~ and ends at phase j E E+ when the amount of the aggregate loss first upcrosses the level 0. This explanation can be applied to the CTMMP {{Jt,St)}t>o- Based on the above remark, we have that VE'(O) is just the upcrossing probability matrix r\ of the model defined in [13]. Then, according to Theorem 2 of Asmussen et al. [13], we have the following result regarding to ^(0). Theorem 5.3.2 M/(0) satisfies the following algebraic Riccati equation: *(0) 2<+_)¥(0) + *(0) I<++) + I<~ >¥(0) + I<_+) = 0. (5.7) To solve Equation (5.7) for *(0); take 5 to be large enough such that 5 > — [T ]iit for all i E E~~, then the solution \l/(0) of (5.7) can be computed by the iteration scheme *(0) = lim^oo//™) where r]^ = 0, Proof From Remark 5.3.1, we know that [\&(0)],j is the probability that the process {St}t>o first returns to 0 occurs while the underlying process is in phase j, starting 91 from the decreasing phase i. By Theorem 2 of [13], we have *(0)U + T(—)*(0) + T(-+) = 0, (5.9) where U = T(++) + T(+_) *(0). Substituting the expression for U into (5.9) yields tf (0)T(+->#(0) + *(0)T(++) + T(-_)*(0) + T(-+) = 0. For the second assertion, the proof for the iteration algorithm can be found in Theorem 3.2 of Asmussen [10]. • Theorem 5.3.3 For any s > 0, ^f(s) satisfies the following matrix equation: ¥(s) Z<++) + !<-+> + (I<~) - s A;^)*(S) + *(s) ^"^(s) = 0. (5.10) 1 Proof Let P0 be the probability matrix that the composite CTMMP {((Jf , Jt), St)}t>o defined in Section 5.2.1 ever revisits 0. From Proposition 5.2.1, for i € E~, j G E+, we have \Pohj = Hn = Ez[I{Tl sri = ET1[e" • I{jn=j}| Jo = i] = [*(s)]y • 92 Applying Theorem 5.3.2 to the CTMMP {((Jf', Jt), St)}t>0 yields +) P0(T(++) + T^+^Po) + (T(—> -sA^)P0 + T(" = 0 . Since P0 = *(s), the above equation is equivalent to (5.10), which establishes the theorem. • The following results about \P(s) and T(s,u), which are proved in the Ap pendix, show that both matrix functions are smooth in [0, oo). Theorem 5.3.4 V&(s) is a smooth matrix function for s E [0, oo). Theorem 5.3.5 T(s,u) is a smooth matrix function for s E [0, oo). Corollary 5.3.6 For any u > 0, the Laplace-Stieljes transform G(s,u) of the time of ruin is a smooth matrix function in s, s E [0,oo). Therefore, the random variable, the time of ruin, has any order moments. 5.4 Moments of the Time of Ruin In this section, we derive formulas for calculating the nth moment matrix of the time of ruin (n E «/K+), which is given by n) n /4 = (-l) ^G(S,u) s=0 dn = (-i)"d^(*MrM) s=0 where the derivative of a matrix is defined entry-wise. Before proceeding to derive the moments matrix fir , we first give some results in the next section for later reference. 93 5.4.1 Preliminaries Since for two matrix functions A(t) and B(i) with same dimensions, even when A(t) = B(i), the multiplication of ^A(t) and B(i) does not usually commute. Because of this, the following results are useful in our derivation. We assume that all matrix multiplications in the following results are conformable. Lemma 5.4.1 For n £ ^V+. If matrices Aft) and B(t) both have nth derivatives w.r.t. t, then k=0 V / Corollary 5.4.2 For n E JV+. If matrices A(t) and B(t) both have nth deriva tives w.r.t. t, and C and D are scalar matrices, then 1. |^(t) • C B(t)) = ± (*) (^k>(t) • C- B^(t)) 2. a D + tC)A{t) = nC-A{a~1)(t) + (D +tC) • A{a)(t) dtn . The following result is useful for solving linear matrix equations, which can be found in Graham [34]. Proposition 5.4.3 The matrix equation AX+XB=C (5.11) has a unique solution X if and only if matrices —A and B have no eigenvalue in 94 common. In this case, the unique solution is given by r i_1 vec(X) = A®B -vec(C), where vec is a vector operator defined in Section 4-4-2. The following two results are related to solving matrix equation (5.10), which will be used in the derivation of the moments expressions. Theorem 5.4.4 The matrix equation i++) { ) 1 i+ ) (+ ) XT + T — X-A^ _)y(0) + Xl - y(0) + V(0)T - X=0 (5.12) has a unique solution which is given by vec(X) (5 Proof First, rewrite Equation (5.12) as ++ +- + } x(V ) + T< >*(O)) + (T<—> + *(O)T( - )X = A-(ij*(o). From Proposition 5.4.3, this equation has a unique solution if and only if — ( T^++'+ T(+")*(0)) and T(-_) + ¥(0)T(+-) have no eigenvalue in common, which is also equivalent to (T(++) + T(+_)*(0)) © fat—) + *(0)T(+-)) is nonsingular. Since \&(0) can be explained as the "upcrossing phase probability" matrix for ( ) (+ ) CTMMP {(Jt, St)}t>o (recall Remark 5.3.1), T " +*(0)T " is equivalent to the K matrix defined by Ramaswani in paper [47] and T(++) +T(+-)*(0) is equivalent 95 to the U matrix defined by da Silva Soares &; Latouche in paper [21] (both matrices were introduced in Chapter 2). The only thing we need to point out is that here we consider the upward first passage time of {St}t>o for the U, while da Silva Soares & Latouche considered downward records (see [21], P. 97-98). A similar situation exists for matrix K. Thus, the eigenvalues of matrices (T(++)+T(+_)*(0)) and (T(~_)+*(0)T(+-)) are all located in the left half of the complex plane. And also from Proposition (++) (+_) 2.3.2, ('T(++)+T(+")*(0)) is non-singular. Therefore-(T +T tf(0)) and (T(—> + #(0)T(+-)) have no eigenvalue in common, which means that Equation (5.12) has a unique solution. The derivation of Equation (5.13) can be found in Graham [34]. • 5.4.2 Computational Procedures Since the nth moment matrix of the time of ruin is given by n ^(l)H-ir^(*(*)r(^)d ) it follows from Lemma 5.4.1 that (n) #*% = (-irE^)(* (o)-r^)(0/u)\. (5.14) fc=0l—a \ / Thus, the next job is to establish procedures to evaluate ^^(O) and r^(0,«), for 0 < k < n. In this section, we first derive the formulas for computing the moments of the time of ruin by assuming all conditions hold and the theorems guaranteeing these operations will be given in the next section. 96 Calculation of *w(0), k G JV We start with *(0)(0). For k = 0, *(0)(0) is just *(0), which turns out to be the solution of Equation (5.7). We can use the algorithm described in Theorem 5.3.2 to obtain it. Suppose now that we have obtained \I/' _1^(0), for k € JY+ . For the terms in Equation (5.10), from Corollary 5.4.2, we have the followings. ^(*(s)T(++)) - *«(s)T(++), k k + k -0 dskl'*(s)T< ->¥(s)) = J2 ( ) (^MT^*** ^)) . i-0 ^ ' In the sequel, we denote QW^^-'-SA;,1.,)^), (5.15) then, 1 ^Q(s) = (T<~> - s Ar"( _))*W(S) - kA^V^s). (5.16) Thus, taking the kth derivative w.r.t. s for each term in Equation (5.10) yields 1 fe 1 ^M(S)T(++) + /T(—) _ s A;(M*W(s) - /fcA;( _)*( - )(s) +E(fc)(*w(s)T(+")*(fc"i)(s)) = o. i=0 97 Rearranging the foregoing expression, we have k (i) kA-^^is) - Y, ( ) (* (s)T<+-> *(*"*> (s) Since fT(++) + T(+_)¥(s)) © ^T(__) - s A"^ + *(s)T(+_)) is nonsingular and by employing Proposition 5.4.3, we get \|/(fc)( (++) + ) ) vec (T + T<- - *(s)y © (T(~> - s A^i, + *(s)T(+- ) fc-i 1 1 vec(fcA;( _)*(*- )(s) " E ( • ) {^(s^-W^is i=l (5.17) Since ^f(s) is smooth on [0, oo), we may take s = 0 in above equation and get (fc) T (+ ) vec(* (0) T(++) + T(+-)*(0)) © (T<—> + *(0)T - )1 feA^^O) - J2 (k) (*(i)(0)T(+-)*(fc-i)(0))) Thus, for any n 6 ^+, \E^n'(0) can be recursively computed with ^(0) as the initial value. We form these results in the following theorem. Theorem 5.4.5 For any n € jV, the nth derivative of SE^n'(0) is given by the following formulas. For n = 0, >P^(0) = ^(0) satisfies the following algebraic Riccati matrix equation: *(0)I<+-)*(0) + *(0)I<++) + I<—)*(0) + ^_+) = 0 98 and *&(0) can be obtained by the algorithm described in Theorem 5.3.2. For 0 < k < n, ++) + r ) <+ ) vec *«(0)j = [(V + I< -)*(0)) ©(2<— + *(0)2 - fe-i /7 vecjW^^^O) - £ ^] (*«(()) Z^-^-^O))) . (5.18) Calculation of rw(0,u), k ^ JV From Theorem 5.3.1, r(s,n) = eH(s)u. Expanding exponential form en^u yields (5.19) <=o z' The method used to evaluate I^ '(0,u) is directly to take derivatives w.r.t. s for the infinite summation. For the case of the first derivative, we have d r(S,«) = E|im») ds j=i Letting s = 0 yields u'/d (5.20) s=0 s=0 99 Since H(s) = T(++) + T(+-}*(s). (5.21) we have H(1)(0) = T(+-}*(1)(0). For i > 2, multiplying each term in Equation (5.21) by ET_1(s) to the right, we get H'(s) = T^IT-^s) + T(+")*(s)ff-1(s), then taking derivatives of the terms in above equation w.r.t. s and letting 5 = 0 yield = H(0)^Hi-1(s) + T^-^^Hoj-^-IT-^s) (5.22) i^ s=o ds ds s=0 Then, the first moment matrix of the time of ruin can be calculated by the following formula (1) (1) (1) ^Tftl ) - -*(o)-r (o,u)-* (o)-r(o,^). Numerical procedures show that the above method is problemlistic. Here is the 100 r^(0,u) matrix for the fourth model in Example 5.(3.1 with u = 10 and 9 — 0.5. -3.51 x 1022 -4.06 x 1022 6.53 x 1021 2.69 x 1022 -4.45 x 10^ 3.62 x 102d 5.34 x 1022 -1.36 x 10i2 3 r«(o, 1.19 x 102b 4.30 x 102b -8.77 x 1024 1.46 x 10~|2 6 5.53 x 1024 3.98 x 1025 -4.48 x 1024 -5.80 x 10 24 The problem results from the fact that after each iteration, the entries in matrix Hn(s) are getting bigger and bigger as n increases. Some significant digits s=0 have been lost due to the overflow. In this example, the exit tolerance for iteration is 10~15 and iteration number is 327. Due to this reason, we uniformize the CTMC with infinitesimal generator T, which is the time re-scaled process of {Jt}t>o, and then use same procedures used before to take kth derivatives of the resultant infinite matrix sum to obtain kth moment of the time of ruin. Precisely, Let 77 = max{-[T<++>]«} and P(s) = I + -H(s) = I + - (T(++) + T(+-)¥(s)>) . (5.23) r? 77 \ / Since, for each s e [0, 00), ^f(s) is a probability transition matrix, it is obvious that by taking such 77, 0 < [P(s)]y < 1 and P(s)e < e. Then, the expression of 101 T(s,u) can be rewritten as H u u < r(s,u) = e ^ = ^e-" ^-P (5). (5.24) »=o For any k G ^+, from Theorem 5.5.1, we have that summation and kth deriva tive w.r.t. s in (5.24) are exchangeable, so that dsk v ' *-( z! dsk v ; Then, letting s = 0 in the above equation yields fkT(s,u) =Ve-Mip'(S) . (5.25) dsk v ' s=o ^-f %\ dsk w o-s=n0 v y Remarks We observe that factor e_r?u forces the matrix entries e~'?u^2^-^Pl(s) not s=0 too large in each iteration. The other is that all P"(s) are probability matrices and 7? r? n our numerical results show that when k is fixed, the matrices, e~ "^ ^/ jp-P (s) s=0 are non-negative or non-positive iteratively when n increases by 1. This is the second reason why the entries in the resulting matrix T^k'(0,u) keep small after each iteration. for all i, k G JV+ in order to So far, the left work is to calculate ^P'(s) s=0 obtain all moments of the time of ruin. To this end, we consider the following cases. Case 1: i = 1, k € J/'+. 102 Taking kth derivative w.r.t. s on both sides of (5.23), we get pC'(s) = -T<+->*W(s), (5.26) V and then letting s = 0 yields p(*)(0) = -T'+-»*'*'(0). (5.27) V Case 2: i > 2, k In this case, multiplying each term in (5.23) by Pl_1(s) to the right yields the following equation P'(s) = fl + -T^P^fs) + -T^-)^(s)Pi-\s) then, taking kth derivative of each term w.r.t. s in above equation, we get ++ + i 1 = fl + -T( ) + -T( -)*(S))^P - (S) = ^)^'w+^.£Q(^.(.))(^(.)), 103 i.e., k 1„^^4 A\/d -J 5 28 dsk p-(S) - pwip»w+i*-> E (&q»w) (&•"«) -' ' ' i s« By letting s = 0 in Equation (5.28), we have F P k P\s) ds s=0 ^ 'M,^ W ,W ^|C)(^* )L(^ ) s=0 We group the results above into the following theorem. Theorem 5.4.6 For any k E Jf+, the kth derivative ofT(s,u) w.r.t. s evaluated at s = 0 is given by dk_ , ^ ^(^)'/dk k k ^M-Hds «=o *—< i\ Vds ' s=0 w/iere 77 = maxj{—[1^ ]jj}, and jor-P^s) for i € jV+ can be obtained by the ls=0 following procedures: 1. For i=l: p(fc)(0) = il<+-)*W(0) (5.29) V 104 2. For i > 2; k w k v J ^ s=o ds \s=o 1 „^_> 5A A\ / dk"J ^-EQ^-OL^e s=0 (5.30) 5.5 Convergence of the Algorithm In this section, we establish a theorem of the convergence of the algorithm given in the previous section. To this end, we first recall some concepts of matrix calculus. We assume all matrices considered here have dimension kx k. The convergence of sequences of matrices is defined entry-wise, i.e., if {An} is a sequence of matrices, An —s- 0 as n —> oo means that [An]y —» 0 for i,j = 1, 2,..., k. For a matrix A, define a matrix norm ||-|| as, ||A|| = Ylij=\2 fcllAlul- Then it is obvious that || A|| = 0 if and only if A = 0 and the convergence A„ —>• 0 as n —> oo is equivalent to the norm convergence: ||A„|| —> 0 as n —>• oo. Furthermore, for a number a and matrices A and B, ||A + B|| < ||A|| + ||B|| (triangle inequality) ||AB|| < ||A|| ||B|| \\aA|| = \a\ IIAII . The following result guarantees that the algorithm given in Theorem 5.4.6 is convergent. 105 Theorem 5.5.1 Let SQ be a fixed positive number and n,k G |jP"(s) are given by (5.26), and for n > 2, ^P"(s) are given by (5.28), then the infinite summation of matrix function d* (5.31) dskJ*<•.« > = £«-*?(£*"<•>) is uniformly convergent on [0, SQ\. Proof For 1 < i < k, let •*jm..i ~~~ •P»( ) max ds S 0 Since ^l[s) is smooth on [0, oo), ip is finite. From (5.26), for 0 < i < k, Pw(s) = -T^-^Vs) V we have maxo<, a = max{l, b, v} < oo . The remainder of the proof is separated into several steps. Step 1. For k = 1, Taking matrix norm on both sides of (5.26) and applying the triangle inequality, 106 we have T(+- 1 p,(s) = P - («)) ! w|p"w 7/ -(s'w)^ < p(S)ip"(S) + ix<-> (1»W) (P«(.)) p :-i < IIPWII- + •IIPWII l "(»» ?7 as < a- + a< ^p' -'w ) i.e., Pl(s] Taking the maximum among s G [0, SQ] on the right hand side of above inequal ity and then on the left yield max p < a • max P-^s) + a\ s '<" 0 i.e., % 1 l xiti < a Xj_i,i + a < ... < a x^^ + (i — 1) a < ia . Thus, there is a polynomial /{(i) '{i)/ with order 1 such that x%,\ < A(i) / (i) -a (5.32) 107 Step 2. Assume that for j = 1, 2,..., k — 1, there exists a sequence of polynomials ti2\ti3\---jti1} such that U)r xit] where the superscript of j\ denotes the order of the polynomial j\ . We now prove that there exists a polynomial f^ with order at most k such that %i,k Using the same procedures as in step 1, we have k fc-i d • # * P(s,iLp.-.(s, + iT«+-.E(«)(-*(s))(|Ip.-, i=i dk < IIPOOI fc-i iTi+-)^i*(»k ) v ; +E r] dsk~-J v ; dsJ J=I fc-i < a dk -go £'"» thus, dk fc-i max it P'to < a • max max v y k dsJ 0 fe-i Xi,k < axt-i,k + a • Y^ ( . ) • Xi-i •53 0 fc-1 < ax^ + a'-J^^Yf^ii) by (5.33) Let f(i) = X)i=i CO ' fjv)i then it is obvious that the order of the polynomial / is at most k — 1. Thus, 1 xi 1 < a (a Xi_2>fc + a'- • f(i - 1)) + a2 Xi-2,k + a* (/(i - 1) + /(i)) 1 < a'" xlifcW-(/(2) + /(3)+ ... + /<£ i i < a + a -(f(2) + f(3) + ... + f(i)) = a* • (l + /(2) + 7(3) + • • • + /(i)) Let /fcfc)(0 = ! + 7(2) + 7(3) + ... + f(i), then the order of the polynomial /f) is at most k and k) xhk Therefore. n -TJU (77 u) max k p-(s; n! \Q fe Since the order of the polynomial fy( ) i^s at most k, we have that -riu(j^r.fw{n) By the uniform convergence theorem of a sequence of functions on a closed in terval (see Theorem 7.12 in Rudin [53]), the above inequality implies that e~vu- X^Li n! ( ^k^>"(s)) ^s uniformly convergent on the interval [0, SQ\. • Corollary 5.5.2 One uniform upperbound of J2™=i ^r (&F ^(s)), for s e [0, s0], is ^wen fry fl )«(l« )° ,(k), E< n! #»• n=l Therefore, for any e > 0, we may choose an integer N such that E e^^L.ff){n) then 00 A^ k < e J2 ds^--ris)' s= 0 n=iV+l 110 and kJ w k v ds s=o ^ n! ds s=0 n=0 if e is small enough. 5.6 Examples Example 5.6.1 In this example, we reproduce the numerical results in Drekic &; Willmot [26]. The models are all classical with inter-arrival rates being 1 and claim sizes being phase-type. The first four standardized moments (mean, coefficient of variation, coefficient of skewness and coefficient of kurtosis) of the time of ruin conditioned on ruin occurs were computed. The four claim size distributions which all have mean 1 are listed below and the numerical results are shown in their Table 1. Exponential Claim Size P(x) = 1 - e~x, x>0. Erlang-6 Claim Size P{x) = 1 - e^Y, ' x>0. j=0 J! Combination of Two Exp. Claim Size P(x) = 1 + -e'6x - -e~6x/5, x>0. v ' 4 4 ~ Ill Mixture of Two Erlangs Claim Size P(x) = 1 - -^ - ~e~^5 - ~xe-9* - ~xe~3x/5, x>0. Mixture of Three Exponentials P(x) = 1-0.0039793e-°014631a;-0.1078392e-°-190206a:-0.8881815e"5-51451, x > 0. Table 5.1: Conditional standardized moments of the time of ruin 0 = 0.1 0 = 0.5 u mean cv skew kuri mean cv skew kurt Exponential Claim Size 0 10.00 4.58 13.74 317.57 2.00 2.24 6.62 76.20 10 100.91 1.47 4.24 32.92 15.33 0.87 2.23 11.25 30 282.73 0.88 2.53 13.65 42.00 0.53 1.34 5.97 Erlang-6 Claim Size 0 5.83 4.53 13.74 317.66 1.17 2.13 6.63 76.45 10 97.37 1.13 3.29 21.04 15.67 0.63 1.68 7.67 30 283.51 0.66 1.93 9.19 45.46 0.37 0.98 4.61 Combination of Two Exp. Claim Size 0 8.61 4.58 13.74 317.59 1.72 2.22 6.62 76.26 10 99.51 1.38 3.96 29.10 15.14 0.81 2.08 10.17 30 281.77 0.82 2.35 12.20 42.07 0.49 1.24 5.55 Mixture of Two Erlangs Clair n Size 0 21.11 4.59 13.73 317.40 4.22 2.25 6.58 75.55 10 112.28 2.03 5.85 60.00 17.48 1.18 3.07 18.62 30 293.62 1.26 3.61 24.69 44.26 0.75 1.92 9.07 Mixture of Three Exp. Clain I Size 0 215.99 4.95 13.79 319.35 43.20 2.92 6.94 81.47 10 525.57 3.13 8.95 136.67 103.94 1.80 4.55 37.45 30 831.40 2.47 7.16 88.51 155.07 1.44 3.81 27.22 From Table 5.1, comparing with Table 1 in [26], we know that all values are the 112 same. We report that by making use of the recursive formulas, the running time for four moments is less than 30 seconds when running on an IBM T30 Laptop. Example 5.6.2 In the second example, we also compute the first four standard ized moments of the time of ruin for the contagion example introduced in Section 3.2.3. The parameters are given in Table 5.2 and the initial probability vector for the environment process is 8+ = {0.5,0.5}. The computed results are presented in Table 5.3. Table 5.2: Parameters used in the model State 1 {aA = 0.02) 2 (aB = 1) Claim rate 1 10 Mean of claim size l/lix = 0.2 1/^2 = 3 Premium (c) 1 1 Table 5.3: Conditional standardized moments of the time of ruin u mean cv skew kurt "1 84.92 3^53 9/78 162.60 10 116.85 3.04 8.28 117.27 30 186.92 2.42 6.50 73.36 We next use the difference quotient of G(s) to approximate the mean. G{h) - G(0) ^ s=0 h where h is a small number. The approximated conditional standardized means are shown in Table 5.4 with various choices of s. Our findings from Table 5.4 are: (1) All values are less than 113 Table 5.4: Approximated conditional standardized mean of the time of ruin u io-2 10"5 10~7 1 70.85 84.82 84.92 10 78.98 116.56 116.85 30 94.36 186.17 186.91 the corresponding ones in Table 5.3. This is because the LST of the time of ruin is convex and decreasing near s = 0; (2) When h is smaller (closer to 0), the approximation is better. 5.7 Mean Busy Period of Perturbed Fluid Flow Models In this last section of the chapter, we study the mean busy period of the perturbed fluid flow model in a similar fashion as that for the moments of the time of ruin. Again, we start with establishing the LST of the busy period of the model. 5.7.1 Laplace Transform of the Busy Period The Laplace transform matrix of the busy period P(u) is defined by the elements sP [®{s)]tj:=^{uA[e- ^l{Jp{u)=j}}, i,jeE and s > 0, where E(Uji) is the conditional expectation when the initial flow level is u and the initial state J0 = i. Specifically, 0(0) is the probability transition matrix of the flow level down from level utoO. The following result regards to the Laplace transform of the busy period of the 114 perturbed Markov-modulated fluid flow model. Theorem 5.7.1 Assume s > 0. The Laplace transform matrix &(s) of the busy period of the fluid flow process is given by G(s) = eu^u, where matrix U(s) satisfies the following quadratic matrix equation 2 A((T2/2) U {s) + Ar U(s) +A-sI=0. (5.34) Furthermore, the following iteration scheme can be used to solve (5.34) for U(s) n l n (0 Ui + \s) = ^3(U<- \s)) with U \s) = -AUJ, (5.35) where matrix function ^(-) is defined by 1 i;3(V) = {A(2WCT2) + A(2AT2)(A - sI)}(VI- V)- - Aw and formulas to calculate parameters fj,, u> and rj are given by the following for mulas: n . rf 2/j.i n /r? 2fii and j] is chosen so large that /i,; > A, + s for all i € E. To prove this theorem, we also need a result regarding the first passage time to level 0 from u. For this purpose, we reflect each sample path of {Vt}t>o about 115 the line x = u Vt = 2u-Vt. It is easily seen that {Vt}t>o is also a Markov-modulated flow level process defined as the net rate in state % is equal to — r* and other parameters, infinitesimal variance of, remain same, and the modulator process is still {Jt}t>o- Consider a particular sample path of {Vt}t>o with first passage to the level 0 from the level u, then there is one and only one sample path from {Vt}t>o which is symmetric to that from {I4}t>o about the line x = u and first upcrosses the level 2u at the same time instant. Since this correspondence is one-to-one, the probability of the process {Vt}t>0 first passage to the level zero is equal to the probability that {Vt}t>o first upcrosses the level 2u both from their initial level u. One such example is shown in Figure 5.2. If we denote the generator of the life time maximum for the process {t^} as U, then we have the following. Theorem 5.7.2 The probability P0 of the flow process {Vt}t>o first down to the level zero is 0u P0 = e , (5.36) where u is the initial flow level and matrix U is the generator of the life time max tli imum of the reflected process of {Vt}t>o- The (i,j) entry of PQ is the probability that the flow starts from state i and depletes at state j. Thus, we are now in the position to prove Theorem 5.7.1. 116 Figure 5.2: One-to-one relationship of sample paths between {Vt}t>0 and {V^}t>o. (red: the one crossing the bottom line is the reflected version) Proof of Theorem 5.7.1 Let \Po]ij be the probability that the fluid process {Vt} first visits fluid level 0 from the initial level u, starting at phase i, before an exponential time period Hs is completed, ending at phase is j. Then by similar reasoning as in the proof of Theorem 5.3.3, we have that Pol [©(*)]« which means that the LST of the busy period P(u) is the same as the probability matrix of the flow going down to level 0 before an exponential time horizon. Also, by Theorem 5.7.2, we have that =U(S)« 117 Thus, we have 0(s)=eu^". To prove formula (5.34), we know that, from Section (3.3.1), U(s) is a fixed point of function tp3, i.e., 1 U(s) = {A(2^/CT2) + A(2/CT2)(A - 5l)}(»7l - U(s))" - Aw . Multiplying each term on both sides of this equation to the right, we get 2 r/U(s) - U (s) = A(2MAT2) + A(2/a2)(A - si) - A,u + AwU(s). Rearranging it and noticing that r) — LOi = —2rj/cr2 and rju>i = 2 fii/crf yields 2 0 = U (s) + A(2M/O.2) + A(2/ 2 = U (s) + A(2/CT2)(A - si) + A_r?IU(s) 2 = U (S) + A(2/ The iteration algorithm (5.35) can be obtained again from Section (3.3.1). • 5.7.2 The Mean Busy Period We next consider the mean busy period of the fluid flow. Since rv(s)u fU"-U"(g) 2^ n.\ 118 we have P = >(*)] = e V(s)u] ~:r s=0 n n °° u -fsXJ=o (s) as = -E s=0 ra=0 n! We next evaluate u"W for n G ^/K"1 ds s=0 We know that U(s) satisfies Equation (5.34) and U(0) can be obtained from the algorithm described in Theorem 5.7.1. Differentiating both sides of (5.34) w.r.t. s, we have (1) Aay2 • (U«(s) • U(s) + U(s) • UW(«)) + Ar • U (s) -1 = 0. Letting s = 0 and rewriting the above equation as follows: A„2/2 • U«(0) • U(0) + (ACT2/2 • U(0) + Ar) • U«(0) -1 = 1, (5.37) then applying vec operation on both sides of above equation, we have that T (1) (1) U(0) <8> A^/a] • vec(u (0)) + [i vec (I) 119 i.e. (1) T vec (U (0)) = [i ® (Aff2/2 • U(0) + Ar) + U(0) 2 U (s) = -A2r/(T2 U(s) - A2/(r2 • A + sA2/c (5.39) and differentiating both sides of it w.r.t. s, we have 2 U (S) - A 2 -gU(s) + A .2 ds 2r/ Then A^2r/ For n > 3, by multiplying (5.39) to the right and differentiating it with respect to s, we have 1 ^2r/cr2 -U™" ^ Ay^-A.-lJ^(s) ^ s=0 ds s=0 s=0 +A2/CT2-U^(0). (5.41) Algorithm 5.1 1. U(0): First, we know that U(s) satisfies Equation (5.34) and then U(0) can be obtained from the algorithm described in Theorem 5.7.1 by letting s = 0; 120 2. U(1)(0): The U(1)(0) is given by Equation (5.37); 2 3- £U (S)|S=0: A2I./ n 4. ^U (s)|s=0forn>3: n 2 U (s) ^/^ rU"">) - A2/CT2 • A • TU"" (S) ds' s=0 ' ds U=o ' ds s=0 n 2 +A2/ • In principle, we can have higher moments of the busy period by taking higher derivatives of the LST given in Theorem 5.7.1 w.r.t. s and letting s = 0. A similar result as Theorem 5.5.1 guarantees that the algorithm is convergent, which is the next theorem. The proof of it is given in the Appendix. Theorem 5.7.3 Forn € Jf+, ^lT{s) is given by the algorithm 5.1, then the s=0 infinite matrix summation n u / d jjn to) s=0 n=l n! Vds is absolutely convergent. Finally, we present a numerical example which shows that it is stable. The reason may be that the coefficient matrices are simple, a diagonal matrix or a 121 product of a diagonal matrix and a generator. (If needed, a uniformization scheme could be applied.) Example 5.7.1 Assume that the environmental phase process has two states de noted as 1 and 2, the transition rates are A12 = 9 and A21 =21, the linear trends are r\ = 0 ri = —8, and the infinitesimal variances are a\ = a\ = 4. ' 2.11574 2.10648 P = 2.06018 2.05092 \ The iteration number is 109, computing time is 23 seconds and exit tolerance is 10-10. We next use the difference quotient of 0(s) to approximate the first derivative of it when h is taken to be a very small positive number: 0(/i) - 0(0) 0(s) ds s=0 h We used s — 0.0001 and the approximated mean busy period is show below. ' 2.11520 2.10595 A = 2.05967 2.05042 We can see that two results are very close and each entry in P is bigger than the corresponding one in P\, this is because each entry of 0(s) is convex near 0. 122 5.8 Appendix Proof of Theorem 5.3.4 We first prove \J/(s) is continuous in [0, oo). For fixed SQ G [0, OO), by definition of *(s), each entry of it is a decreasing function of s and bounded by the corre sponding entry of \&(0), so by the Lebesgue's dominated convergence theorem (see Chung, [20]), for s G [0,oo), _5n lim [¥(s)]ij = lim Ei[e ][{ri«X)i j =j}] ST1 = Ei[lim e' I{Tl = [*(so)]ij, which means that *(s) is continuous on [0, oo). We now use mathematical induction to prove \&(s) is smooth on [0, oo). For any s,s0 G [0, oo) and s ^ s0, evaluating Equation (5.10) at s and s0, respectively, and taking the difference of these two equations, we have ++ *(S) - *(SO))T( ) + [(T(~> -SA;^(S) - (T(-) - So A^))*^) (+ ) (+ ) + [*(s)T - *(s) - *(s0)T - *(s0)l = 0. (5.44) Denote s - s0 123 Then, Equation (5.44) can be rewritten as ++) + ( ) (+ ) 1 X(s)(V + T< ->*(s0)) + (T — + *(s)T - - sA- _))x(s) V-) *(*>)• (5-45) Thus, the solution X(s) of Equation (5.45) can be expressed as -l + + vec (X(s)) = [(T( +) + T( ->*(so)) e (T<—> + *(s)T<+-> - sA;^' •Ar"(-)*(*o). It is obvious that lim vec(X(s) exists and 1 lim vec(X(s) T(++)+T(+_)^(so)^0^T(__)+^r(so)_SOA_^jj- A_i^(so) S^SQ which means that *(s) has first derivative at So- Since s0 is any point in [0, oo), we have that *(s) is continuously differentiable on [0, oo). Suppose now that \P(s) has up to kth continuous derivatives, k € JV+. We now prove ^f(s) has (k + l)th derivative and therefore *&(s) is a smooth matrix function on [0, oo). Taking kth derivative w.r.t. s in (5.10) yields (fc) 1 #M(S)T(++) + fT(—) _ s A^))* (s) - JtA^,*^- ^) k + fc + J] ( ) (*W(5)T( -)*( -«)(s)) = 0 , i=0 ^ ' 124 which is equivalent to #(fc)(s)(T(++) + T^-^s)) + fT(—} - s A-ji, + *(s)T(+-n*^(s) = fcA^,***"1^) - ]£ (*) ^(s^-W^is)) . From Proposition 5.4.3, above equation has a unique solution \J/' \s) and it can be expressed as (fc) (++ + ( ) 1 (+ ) vec(* (s)) = (T ) + T( -)*(S)) ®(T — -SA;( _) + *(S)T - fc-i 1 fc 1 vec(fcA- „)*( - )(S) - J2 J^^jT^*^^))) . (5.46) By the assumption that *(s) has up to kth continuous derivatives on [0, oo), then the right hand side of (5.46) has continuous derivative, which means that each entry of vecf tyk\s) J has continuous derivative on [0, oo). So, \&(s) has (k + l)th continuous derivative. By mathematical induction, \&(s) is smooth on [0, oo). • Proof of Theorem 5.3.5 Since *(s) is smooth on [0, oo), Pn(s) is smooth on [0, oo) as well, for any integer n. From Theorem 5.5.1, we know that for any integer k, ES(SP"W) n=l r an is uniformly convergent on [0, s0] f° Y fixed s0 > 0. Therefore, -^T(s,u) is a continuous function on [0, oo). Since k is an arbitrary non-negative integer, we have that T(s, u) is a smooth matrix function on [0, oo). • 125 Proof of Theorem 5.7.3 We merely consider summation ES-On! Vds M s=0 n=3 which can be rewritten by plugging in expression (5.43) as v ^ n! Vds ' s=0 n—3 d 2 - A^2/cr9/„2: • A • ^U"- fs ESr V>'( s=0 s=0 n=3 ds ds n 2 + A2/CT2 • U ~ (0) Let Vn(s ds s=0 and a = max] ||^2:r/a3 ||, ||A2/ By taking Euclidean on both sides of (5.43) and applying triangle inequality on the right hand side, then, for n > 3, we have the following inequality: n-l xn < a xn_i + a xn-2 + a Let a + Va2 + 4a a — \/a2 + 4a o = and c = 126 we have n l n 2 xn - c xn_i + b a ~ < b (xn-i - c xn~2 + b a ~ ). (5-47) Notice that b > 0 and c < 0, we have for any n > 2 1 xn — c xn_i + b a""' > 0 . Therefore, iterating inequality (5.47) results in n l n 2 xn — c xn_i + b a ~ < b (x„_i — c xn-2 + b a ~ ) 2 n 3 < b (xn^2-cxn-3 + ba - ) n 2 < b ~ {x2 -cxi + ba), which is just n l n 2 xn — cxn-\ + ba ~ < b ~ (x2 — cx\+ba). Again, since c < 0, b > 0 and xn > 0, we have n 2 n 2 xn < b ~ (x2 - cxi + da) := e0 b ~ . Thus, oo n < V ^-eo bn~2 < oo . (5.48) E n! &™) s=0 ^—' n! n=3 127 We have thus proved the theorem. • 128 Chapter 6 CONCLUSIONS AND FURTHER WORK This thesis has applied matrix analytic methods to analyze various risk models and perturbed fluid flow models. The risk models that can be considered are quite general in that the methodologies developed in the thesis are applicable to all risk models which have sample-path-equivalent fluid flow counterparts, which include the (perturbed) Sparre-Andersen models with inter-claim time and claim size both being phase-type, the (perturbed) Markov-modulated Poisson risk models with claim sizes being phase-type and the (perturbed) processes with inter-claim time being MAP and claim sizes being phase—type. All these phase—type and Markovian assumptions are required for the existence of the sample-path-equivalent fluid flow, and it is the only limitation for the methodology we presented in the thesis. As pointed out in previous chapters, matrix analytic methods have obtained great success in the literatures of queueing theory and risk theory. The work done in this thesis extends the boundaries of problems that can be solved in risk theory. The risk models considered in Chapters 3 and 4 are perturbed aggregate loss models. The novelty of the work in these two chapters is the development of the approaches to obtaining ruin time related quantities in a unified, tractable way. 129 All these algorithms are readily programmable. Chapter 3 represents the first time to apply Asmussen's methods to calculate the ultimate ruin probability for perturbed risk models. In this method, a sample- path-equivalent fluid flow is first constructed and then the algorithm for ruin probability is built up based on Asmussen's work. In Chapter 4, the Erlangization approach was used to approximate the finite time ruin probability by first identifying a composite fluid flow process which is a combination of Erlang phase and the sample-path-equivalent fluid flow counter part of the risk process. A brute-force algorithm is given to approximate the finite time ruin probability. In addition, we find the block structure of the generator of the fluid flow process and an efficient recursive formula based on such structure was derived. The numerical examples show that it saves running time by several orders of magnitude. In Chapter 5, the moments of the time of ruin were considered in non-perturbed risk processes. The method used for obtaining moments is, to our best believe, new in the literature. We first establish the LST of the time of ruin and then get the moments of it by taking the derivative of the LST with respect to the dummy variable and setting it equal to zero. We obtain the formulae for any order of the moments, which is an infinite sum of matrices. The main issue for this method is stability. To overcome this problem, we use the uniformization technique before taking derivative to the LST. The algorithm is convergent. As for future work, a topic that remains unsolved is the moments of the per turbed risk models, which were discussed in Chapters 3 and 4. Since the sample- path-equivalent fluid flow is partially perturbed, the generator of the underlying Markov process can't fit a single matrix equation, which we have already seen in 130 Chapters 3 and 4. As a result, we could obtain an iteration formula for the deriva tives of the LST. We anticipate that the analysis will be complicated and a precise line of attack is unclear at the present time. The theory and algorithms that have been developed herein to address risk models have applications in other fields of applied probability as well. Recently, Stanford et al. [56] and Woolford et al. [65] consider the applications of the fluid flow theory to the applications for the forest fire perimeter. The containment and escape probabilities of the fire perimeters are obtained by investigating the evolution of an uncontrolled fire perimeter over time. The Erlangization technique was used in their methods. In addition, several first passage time formulas are given as well. However, these papers both assume that the growth in fire size and fire line construction change linearly over time, the perturbed fluid flow model provides us a more flexible setting in this regard. This suggests us investigating the same issues as Stanford et al. [56] and Woolford et al. [65] considered for perturbed fluid flow models. 131 Bibliography [1] ABADIR, K.M. AND MAGNUS, J.R. (2005). Matrix Algebra. Cambridge. [2] AHN, S. AND BADESCU, A. (2007). On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals. In surance: Mathematics and Economics 41, 234-249. [3] AHN, S., BADESCU, A. AND RAMASWAMI, V. (2007). Time Dependent Analysis of Finite Buffer Fluid Flows and Risk Models with a Dividend Barrier. Queueing Systems 55, 207-222. [4] AHN, S., JEON, J. AND RAMASWAMI, V. (2005). Steady State Analysis of Finite Fluid Flow Models Using Finite QBDs. Queueing Systems 49, 223- 259. [5] AHN, S. AND RAMASWAMI, V. (2003). Fluid flow models and queues - A connection by Stochastic copuling. Stochastic Models 19, 325-348. [6] AHN, S. AND RAMASWAMI, V. (2004). Transient Analysis of Fluid Flow Models via Stochastic Coupling to a Queue. Stochastic Models 20, 71-101. [7] AHN, S. AND RAMASWAMI, V. (2005). Efficient algorithms for transient analysis of stochastic fluid flow models. Journal of Applied Probability 42, 531-549. 132 [8] ALBRECHER, H. AND BOXMA, O.J. (2005). On the discounted penalty function in a Markov-dependent risk model. Insurance: Mathematics and Economics 37, 650-672. [9] ASMUSSEN, S. (1994). Busy period analysis, rear events and transient be havior in fluid flow models. Journal of Applied Mathematics and Stochastic Analysis 7, 269-299. [10] ASMUSSEN, S. (1995a). Stationary distributions for fluid flow models with or without Brownian noise. Commum. Statist. - Stochastic Models 11, 21-49. [11] ASMUSSEN, S. (1995b). Stationary distributions via first passage times. Ad vances in Queueing: Models, Methods & Problems (Dshalalow, J ed.), 79- 102. CRC Press, Boca Raton, Florida. [12] ASMUSSEN, S. (2000). Ruin Probabilities. World Scientific. [13] ASMUSSEN, S., AVRAM, F. AND USABEL, M. (2002). Erlangian approxi mations for finite-horizon ruin probabilities. Astin Bulletin 32, 267-281. [14] ASMUSSEN, S., NERMAN, O. AND OLSSON, M. (1996). Fitting phase- type distributions via the EM algorithm. Scandinavian Journal Statistics, 23, 419-441. [15] BADESCU, A. AND BREUER, L. (2008). The use of vector-valued martingles in risk theory. To appear in Blatter der DGVFM. [16] BADESCU, A., BREUER, L., DA SILVA SOARES, A., LATOUCHE, G., REMICHE, M.A. AND STANFORD, D.A. (2005a). Risk processes analyzed as fluid queues. Scandinavian Actuarial Journal 2, 127-141. 133 [17] BADESCU, A., BREUER, L., DREKIC, S. LATOUCHE, G. AND STANFORD, D.A (2005b) The surplus prior to ruin and the deficit at ruin for a correlated risk process. Scandinavian Actuarial Journal 6, 433-445. [18] BERTOIN, J. (1996). Levy Processes. Cambridge University Press. [19] BERTOIN, J. (1997). Exponential decay and ergodicity of completely asym metric Levy Processes in a finite interval. Annals of Applied Probability 7, 156-169. [20] CHUNG, K.L. (2001). A Course in Probability Theory. 3thEd., Academic Press. [21] DA SILVA SOARES, A. AND LATOUCHE, G. (2002). Further results on the similarities between fluid queues and QBDs. In Matrix-Analytic Methods, Theory and Applications Latouche, G., Taylor, P.G., Eds.; Proc. of the Fourth International Conference on Matrix Analytic Mthods; World Scientific Pub lishing Co.: Singapore. [22] DICKSON, D.C.M. AND HlPP, C. (2001). On the time to ruin for Erlang(2) risk processes. Insurance: Mathematics and Economics 29, 333-344. [23] DICKSON, D.C.M AND WATERS, H.R. (1991). Recursive calculation of survival probabilities. Astin Bulletin 22, 199-221. [24] S. DREKIC, J.E. STANFORD AND G.E. WILLMOT (2004). Symbolic cal culation of the moments of the time of ruin. Insurance: Mathematics and Economics 34, 109-120. [25] DREKIC, S. AND WILLMOT, G.E. (2003). On the density and moments of the time of ruin with exponential claims. Astin Bulletin 33, 11-21. 134 [26] DREKIC, S. AND WILLMOT, G.E. (2005). On the moments of the time of ruin with applications to phase-type claims. North American Actuarial Journal 9, 17-30. [27] DUFRESNE, F. AND GERBER, H.U (1991). Risk theory for the compound Poison process that is perturbed by diffusion. Insurance: Mathematics and Economics 10, 51-59. [28] FURRER, H.J. AND SCHMIDLI, H. (1994) Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion. Insurance: Mathemat ics and Economics 15, 23-36. [29] GERBER, H.U AND GOOVAERTS, M.J. (1987). On the probability and severity of ruin. Astin Bulletin 17, 151-163. [30] GERBER, HANS U. AND SHIU, ELIAS S.W. (1997). The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. Insurance: Mathematics and Economics 21, 129-137. [31] GERBER, HANS U. AND SHIU, ELIAS S.W. (1998). Onn the time value of ruin. North American Actuarial Journal 2, 48-78. [32] GERBER, HANS U. AND SHUI, ELIAS S.W. (2005a). The time value of ruin in a Sparre Andersen model. North American Actuarial Journal 9, 49-69. [33] GAVER, D.P., LEHOCZKY, J.P. (1982). Performance evaluation of voice/data queueing systems, Applied Probability - Computer Science: The Interface 1, Ott, T.J. Eds. 329-346. [34] GRAHAM, A. (1981). Kronecker Products and Matrix Calculaus with Appli cations. Wiley, NY. 135 [35] JACOBSEN, M. (2005). The time to ruin for a class of Markov additive risk processes with two-sided jumps. Advances in Applied Probability 37, 963-992. [36] KARANDIKAR, R.L. AND KULKARNI, V.G. (1995). Second-order fluid flow model of a data-buffer in random environment, Operations Research 43, 77-88. [37] KENNEDY, J. AND WILLIAMS, D. (1990). Probability factorization of a quadratic matrix polynomial, Mathethical Proceedings of Cambridge Philo sophical Society 107, 591-600. [38] KLEINROCK, L. (1975). Queueing Systems, Vol. 1: Theory Wiley, NY. [39] KLUGMAN, S., PANJER, H. AND WILLMOT, G, (1998). Loss Models - From Data to Decisions, John Wiley, New York. [40] LATOUCHE, G. AND RAMASWAMI, V. (1999). Introduction to Matrix Ana lytic Methods in Stochastic modeling. SIAM. [41] Li, S. AND GARRIDO, J. (2005). The Gerber-Shiu function in a Sparre An dersen risk process perturbed by diffusion. Scandinavian Actuarial Journal 3, 161-186. [42] LIN, X.S. AND WILLMOT, G.E. (1999). Analysis of a defective renewal equation arising in ruin theory. Insurance: Mathematics and Economics 25, 63-84. [43] LIN, X.S. AND WILLMOT, G.E. (2000). The moments of the time of ruin, the surplus before ruin, and the deficit at ruin, Insurance: Mathematics and Economics 27, 19-44. 136 [44] Lu, Y. AND TSAI, C. (2007). The expected discounted penalty at ruin for a Markov-modulated risk process perturbed by diffusion. North American Actuarial Journal 11, 136-152. [45] NEUTS, M.F. (1981). Matrix-Geometric Solutions in Stochtic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, MD. [46] NEUTS, M.F. (1989). Structured Stochastic Matrices of M/G/l Type and Their Applications. Marcel Dekker, New York. [47] RAMASWAMI, V. (1999). Matrix analytic methods for stochastic fluid flows. In Teltrafic Engineering in a Competitive World. Smith, D., Key, P., Eds; Proc. of the 15th International Teletraffic Congress. Elsevier, 1999. 1019— 1030. [48] RAMASWAMI, V. (2006). First passage in fluid flow queues and applications to risk theory. Methodology and Computing in Applied Probability 8, 497-515. [49] RAMASWAMI, V., WOOLFORD, D.G. & STANFORD, D.A. (2008). The Er- langization method for Markovian fluid flows. Annals of Operations Research 160, 215-225. [50] ROGERS, L.C.G. (1994). Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains. The Annals of Applied Probability 4, 390-413. [51] ROLSKI, T., SCHMIDLI, H., SCHMIDT, V. AND TEUGELS, J (1998). Stochastic Processes for Insurance and Finance. Wiley series in probability and statistics. [52] ROSS, S.M. (1996). Stochastic Process. John Wiley & Sons, Inc.. 137 [53] RUDIN, W. (1976). Principles of mathematical analysis, 3d ed. New York: McGraw-Hill. [54] SCHMIDLI, H. (1995). Cramer-Lundberg approximations for ruin probabil ities of risk processes perturbed by diffusion. Insurance: Mathematics and Economics 16, 135-149. [55] STANFORD, D.A., AVRAM, F., BADESCU, A., BREUER, L., DA SILVA SOARES, A. AND LATOUCHE, G. (2005). Phase-type approximations to finite-time ruin probabilities in the Sparre-Andersen and stationary renewal risk models. Astin Bulletin 35, 131-144. [56] STANFORD, D. A., LATOUCHE, G., WOOLFORD, D. G., BOYCHUK, D. AND HUNCHAK, A. (2005). Erlangized fluid flow queues with application to uncontrolled fire perimeter. Stochastic Models 21, 631-642. [57] STANFORD, D.A. AND STROINSKI, K.J. (1994). Recursive method for com puting finite-time ruin probabilies for phase-distributed claim sizes. Astin Bulletin, 24, 235-254. [58] STANFORD, D.A., STROINSKI, K.J. AND LEE, K. (2000). Ruin probabili ties based at claim instants for some non-Poisson claim processes. Insurance: Mathematics and Economics 26, 251-267. [59] SPARRE ANDERSEN E. (1957). On the collective theory of risk in the case of contagion between the claims. Transations of the 15th International Congress of Actuaries New York, II, 219-229. [60] TAKACS, L. Introduction to the Theory of Queues. Oxford Univesity Press, New York, 1962. 138 [61] THORIN, O. AND WIKSTAD, N. (1973). Numerical evaluation of ruin prob abilities. Astin Bulletin VII, 137-153. [62] THORIN, O. AND WIKSTAD, N. (1977). Calculation of ruin probabilities when the claim distribution is lognormal. Astin Bulletin IX, 231-246. [63] TSAI, C.C.-L. AND WILLMOT, G.E. (2002). A generalized defective re newal equation for the surplus process perturbed by diffusion. Insurance: Mathematics and Economics 30, 51-66. [64] WILLMOT, G.E. AND LIN, X.S. (2001). Lundberg Approximations for Com pound Distributions with Insurance Applications. Springer-Verlag, New York. [65] WOOLFORD, D.G., STANFORD. D.A., KULPERGER, R.J., BOYCHUK, D. AND WOTTON, B.M. (2008). Erlangian approximations for the transient analysis of a fluid queue model for forest fire perimeter. To appear in INFOR: Information Systems and Operational Research. [66] YUEN, K.C., YANG, H. AND WANG, R. (2005). On Erlang(2) Risk Process Perturbed by Diffusion. Communications in Statistics - Theory and Methods 34, 2197-2208. 139 Vitae Personal Information Citizenship: Canadian Affiliation: Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario, N6A 5B7 Education Ph.D. (statistics) Department of Statistical and Actuarial Sciences, University of Western Ontario, Canada (Aug 2008) M.Sc. (statistics) Department of Statistical and Actuarial Sciences, University of Western Ontario, Canada (2004) B.Sc. and M.Sc. Department of Computational Mathematics, (Mathematics) Xi'an Jiaotong University, China (1987 and 1990) Employment Sept. 2003-Present Teaching Assistant: Department of Statistical and Actuarial Sciences, University of Western Ontario, Canada May 1999-May 2002 Software Developer: GN Nettest Inc.(Toronto), Canada Sept. 1996-April 1998 Teaching Assistant: Department of Mathematics and Statistics, University of Windsor, Canada Sept. 1993-Aug. 1996 Assistant Professor: Department of Mathematics, Nanjing University of Aeronau tics & Astronautics, China 140 Sept. 1990-Aug. 1993 Lecturer: Department of Mathematics, Nanjing University, China Research 1. Research Interests - Applied probability: stochastic modelling; - Queueing theory and its applications: matrix-analytic methods; - Risk/Ruin theory; - Functional analysis and its applications. 2. Publications Articles in Refereed Journals 1. Kaiqi Yu, David A. Stanford and Jiandong Ren "Erlangian approximation to finite time ruin probabilities in perturbed risk models". Accepted by Scandinavian Actuarial Journal. 2. Serge B. Provost, David A. Stanford and Kaiqi Yu (2008). "On approximat ing the distribution of random distances within and between certain areas and volumes", INFOR: Information Systems and Operational Research. Vol 45, May, 51-63. 3. Kaiqi Yu (1994). "Schrodinger operators with magnetic and electric poten tials", Bull. Austral. Math. Soc, Vol 50, 299-312. Articles Submitted to Refereed Journals 1. Serge B. Provost, Kaiqi Yu and David A. Stanford "On approximating the distribution of distances between random points in two polygons". Submitted to Forest Science; Revision invited (revision intended after submission of thesis). Article in Progress 1. "The moments of the time of ruin in Markovian risk models" (with DA. Stanford and J. Ren). 3. Conference Presentations 1. "Perturbed risk processes analyzed as fluid flows". (joint work with Jiandong Ren, Lothar Breuer and David A. Stanford). CanQueue 2007. Halifax, CA. (August 2007). 141 Computing Skills 1. Strong ability in designing and programming software; 2. Programming languages: C/C++, Java, Fortran 90, TCL/TK, Visual Basic 6, R, S-Plus, Mathematica, SAS. Related Activities 1. Consulting • STATLAB statistical consultant The University of Western Ontario. September 2006 - April 2007 (a) Social Science Help Desk (drop in consulting for Faculty of Social Science) (b) Project work for individual clients 2. Service Positions • Departmental Graduate Student Representative The University of Western Ontario Department of Statistical and Actuarial Sciences (2006-07 & 2007-08 academic years)