Martingales by D
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Stochastic Differential Equations
Stochastic Differential Equations Stefan Geiss November 25, 2020 2 Contents 1 Introduction 5 1.0.1 Wolfgang D¨oblin . .6 1.0.2 Kiyoshi It^o . .7 2 Stochastic processes in continuous time 9 2.1 Some definitions . .9 2.2 Two basic examples of stochastic processes . 15 2.3 Gaussian processes . 17 2.4 Brownian motion . 31 2.5 Stopping and optional times . 36 2.6 A short excursion to Markov processes . 41 3 Stochastic integration 43 3.1 Definition of the stochastic integral . 44 3.2 It^o'sformula . 64 3.3 Proof of Ito^'s formula in a simple case . 76 3.4 For extended reading . 80 3.4.1 Local time . 80 3.4.2 Three-dimensional Brownian motion is transient . 83 4 Stochastic differential equations 87 4.1 What is a stochastic differential equation? . 87 4.2 Strong Uniqueness of SDE's . 90 4.3 Existence of strong solutions of SDE's . 94 4.4 Theorems of L´evyand Girsanov . 98 4.5 Solutions of SDE's by a transformation of drift . 103 4.6 Weak solutions . 105 4.7 The Cox-Ingersoll-Ross SDE . 108 3 4 CONTENTS 4.8 The martingale representation theorem . 116 5 BSDEs 121 5.1 Introduction . 121 5.2 Setting . 122 5.3 A priori estimate . 123 Chapter 1 Introduction One goal of the lecture is to study stochastic differential equations (SDE's). So let us start with a (hopefully) motivating example: Assume that Xt is the share price of a company at time t ≥ 0 where we assume without loss of generality that X0 := 1. -
Poisson Processes Stochastic Processes
Poisson Processes Stochastic Processes UC3M Feb. 2012 Exponential random variables A random variable T has exponential distribution with rate λ > 0 if its probability density function can been written as −λt f (t) = λe 1(0;+1)(t) We summarize the above by T ∼ exp(λ): The cumulative distribution function of a exponential random variable is −λt F (t) = P(T ≤ t) = 1 − e 1(0;+1)(t) And the tail, expectation and variance are P(T > t) = e−λt ; E[T ] = λ−1; and Var(T ) = E[T ] = λ−2 The exponential random variable has the lack of memory property P(T > t + sjT > t) = P(T > s) Exponencial races In what follows, T1;:::; Tn are independent r.v., with Ti ∼ exp(λi ). P1: min(T1;:::; Tn) ∼ exp(λ1 + ··· + λn) . P2 λ1 P(T1 < T2) = λ1 + λ2 P3: λi P(Ti = min(T1;:::; Tn)) = λ1 + ··· + λn P4: If λi = λ and Sn = T1 + ··· + Tn ∼ Γ(n; λ). That is, Sn has probability density function (λs)n−1 f (s) = λe−λs 1 (s) Sn (n − 1)! (0;+1) The Poisson Process as a renewal process Let T1; T2;::: be a sequence of i.i.d. nonnegative r.v. (interarrival times). Define the arrival times Sn = T1 + ··· + Tn if n ≥ 1 and S0 = 0: The process N(t) = maxfn : Sn ≤ tg; is called Renewal Process. If the common distribution of the times is the exponential distribution with rate λ then process is called Poisson Process of with rate λ. Lemma. N(t) ∼ Poisson(λt) and N(t + s) − N(s); t ≥ 0; is a Poisson process independent of N(s); t ≥ 0 The Poisson Process as a L´evy Process A stochastic process fX (t); t ≥ 0g is a L´evyProcess if it verifies the following properties: 1. -
A Stochastic Processes and Martingales
A Stochastic Processes and Martingales A.1 Stochastic Processes Let I be either IINorIR+.Astochastic process on I with state space E is a family of E-valued random variables X = {Xt : t ∈ I}. We only consider examples where E is a Polish space. Suppose for the moment that I =IR+. A stochastic process is called cadlag if its paths t → Xt are right-continuous (a.s.) and its left limits exist at all points. In this book we assume that every stochastic process is cadlag. We say a process is continuous if its paths are continuous. The above conditions are meant to hold with probability 1 and not to hold pathwise. A.2 Filtration and Stopping Times The information available at time t is expressed by a σ-subalgebra Ft ⊂F.An {F ∈ } increasing family of σ-algebras t : t I is called a filtration.IfI =IR+, F F F we call a filtration right-continuous if t+ := s>t s = t. If not stated otherwise, we assume that all filtrations in this book are right-continuous. In many books it is also assumed that the filtration is complete, i.e., F0 contains all IIP-null sets. We do not assume this here because we want to be able to change the measure in Chapter 4. Because the changed measure and IIP will be singular, it would not be possible to extend the new measure to the whole σ-algebra F. A stochastic process X is called Ft-adapted if Xt is Ft-measurable for all t. If it is clear which filtration is used, we just call the process adapted.The {F X } natural filtration t is the smallest right-continuous filtration such that X is adapted. -
Introduction to Stochastic Processes - Lecture Notes (With 33 Illustrations)
Introduction to Stochastic Processes - Lecture Notes (with 33 illustrations) Gordan Žitković Department of Mathematics The University of Texas at Austin Contents 1 Probability review 4 1.1 Random variables . 4 1.2 Countable sets . 5 1.3 Discrete random variables . 5 1.4 Expectation . 7 1.5 Events and probability . 8 1.6 Dependence and independence . 9 1.7 Conditional probability . 10 1.8 Examples . 12 2 Mathematica in 15 min 15 2.1 Basic Syntax . 15 2.2 Numerical Approximation . 16 2.3 Expression Manipulation . 16 2.4 Lists and Functions . 17 2.5 Linear Algebra . 19 2.6 Predefined Constants . 20 2.7 Calculus . 20 2.8 Solving Equations . 22 2.9 Graphics . 22 2.10 Probability Distributions and Simulation . 23 2.11 Help Commands . 24 2.12 Common Mistakes . 25 3 Stochastic Processes 26 3.1 The canonical probability space . 27 3.2 Constructing the Random Walk . 28 3.3 Simulation . 29 3.3.1 Random number generation . 29 3.3.2 Simulation of Random Variables . 30 3.4 Monte Carlo Integration . 33 4 The Simple Random Walk 35 4.1 Construction . 35 4.2 The maximum . 36 1 CONTENTS 5 Generating functions 40 5.1 Definition and first properties . 40 5.2 Convolution and moments . 42 5.3 Random sums and Wald’s identity . 44 6 Random walks - advanced methods 48 6.1 Stopping times . 48 6.2 Wald’s identity II . 50 6.3 The distribution of the first hitting time T1 .......................... 52 6.3.1 A recursive formula . 52 6.3.2 Generating-function approach . -
Non-Local Branching Superprocesses and Some Related Models
Published in: Acta Applicandae Mathematicae 74 (2002), 93–112. Non-local Branching Superprocesses and Some Related Models Donald A. Dawson1 School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Canada K1S 5B6 E-mail: [email protected] Luis G. Gorostiza2 Departamento de Matem´aticas, Centro de Investigaci´ony de Estudios Avanzados, A.P. 14-740, 07000 M´exicoD. F., M´exico E-mail: [email protected] Zenghu Li3 Department of Mathematics, Beijing Normal University, Beijing 100875, P.R. China E-mail: [email protected] Abstract A new formulation of non-local branching superprocesses is given from which we derive as special cases the rebirth, the multitype, the mass- structured, the multilevel and the age-reproduction-structured superpro- cesses and the superprocess-controlled immigration process. This unified treatment simplifies considerably the proof of existence of the old classes of superprocesses and also gives rise to some new ones. AMS Subject Classifications: 60G57, 60J80 Key words and phrases: superprocess, non-local branching, rebirth, mul- titype, mass-structured, multilevel, age-reproduction-structured, superprocess- controlled immigration. 1Supported by an NSERC Research Grant and a Max Planck Award. 2Supported by the CONACYT (Mexico, Grant No. 37130-E). 3Supported by the NNSF (China, Grant No. 10131040). 1 1 Introduction Measure-valued branching processes or superprocesses constitute a rich class of infinite dimensional processes currently under rapid development. Such processes arose in appli- cations as high density limits of branching particle systems; see e.g. Dawson (1992, 1993), Dynkin (1993, 1994), Watanabe (1968). The development of this subject has been stimu- lated from different subjects including branching processes, interacting particle systems, stochastic partial differential equations and non-linear partial differential equations. -
Simulation of Markov Chains
Copyright c 2007 by Karl Sigman 1 Simulating Markov chains Many stochastic processes used for the modeling of financial assets and other systems in engi- neering are Markovian, and this makes it relatively easy to simulate from them. Here we present a brief introduction to the simulation of Markov chains. Our emphasis is on discrete-state chains both in discrete and continuous time, but some examples with a general state space will be discussed too. 1.1 Definition of a Markov chain We shall assume that the state space S of our Markov chain is S = ZZ= f:::; −2; −1; 0; 1; 2;:::g, the integers, or a proper subset of the integers. Typical examples are S = IN = f0; 1; 2 :::g, the non-negative integers, or S = f0; 1; 2 : : : ; ag, or S = {−b; : : : ; 0; 1; 2 : : : ; ag for some integers a; b > 0, in which case the state space is finite. Definition 1.1 A stochastic process fXn : n ≥ 0g is called a Markov chain if for all times n ≥ 0 and all states i0; : : : ; i; j 2 S, P (Xn+1 = jjXn = i; Xn−1 = in−1;:::;X0 = i0) = P (Xn+1 = jjXn = i) (1) = Pij: Pij denotes the probability that the chain, whenever in state i, moves next (one unit of time later) into state j, and is referred to as a one-step transition probability. The square matrix P = (Pij); i; j 2 S; is called the one-step transition matrix, and since when leaving state i the chain must move to one of the states j 2 S, each row sums to one (e.g., forms a probability distribution): For each i X Pij = 1: j2S We are assuming that the transition probabilities do not depend on the time n, and so, in particular, using n = 0 in (1) yields Pij = P (X1 = jjX0 = i): (Formally we are considering only time homogenous MC's meaning that their transition prob- abilities are time-homogenous (time stationary).) The defining property (1) can be described in words as the future is independent of the past given the present state. -
A Representation for Functionals of Superprocesses Via Particle Picture Raisa E
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector stochastic processes and their applications ELSEVIER Stochastic Processes and their Applications 64 (1996) 173-186 A representation for functionals of superprocesses via particle picture Raisa E. Feldman *,‘, Srikanth K. Iyer ‘,* Department of Statistics and Applied Probability, University oj’ Cull~ornia at Santa Barbara, CA 93/06-31/O, USA, and Department of Industrial Enyineeriny and Management. Technion - Israel Institute of’ Technology, Israel Received October 1995; revised May 1996 Abstract A superprocess is a measure valued process arising as the limiting density of an infinite col- lection of particles undergoing branching and diffusion. It can also be defined as a measure valued Markov process with a specified semigroup. Using the latter definition and explicit mo- ment calculations, Dynkin (1988) built multiple integrals for the superprocess. We show that the multiple integrals of the superprocess defined by Dynkin arise as weak limits of linear additive functionals built on the particle system. Keywords: Superprocesses; Additive functionals; Particle system; Multiple integrals; Intersection local time AMS c.lassijication: primary 60555; 60H05; 60580; secondary 60F05; 6OG57; 60Fl7 1. Introduction We study a class of functionals of a superprocess that can be identified as the multiple Wiener-It6 integrals of this measure valued process. More precisely, our objective is to study these via the underlying branching particle system, thus providing means of thinking about the functionals in terms of more simple, intuitive and visualizable objects. This way of looking at multiple integrals of a stochastic process has its roots in the previous work of Adler and Epstein (=Feldman) (1987), Epstein (1989), Adler ( 1989) Feldman and Rachev (1993), Feldman and Krishnakumar ( 1994). -
Lecture 18 : Itō Calculus
Lecture 18 : Itō Calculus 1. Ito's calculus In the previous lecture, we have observed that a sample Brownian path is nowhere differentiable with probability 1. In other words, the differentiation dBt dt does not exist. However, while studying Brownain motions, or when using Brownian motion as a model, the situation of estimating the difference of a function of the type f(Bt) over an infinitesimal time difference occurs quite frequently (suppose that f is a smooth function). To be more precise, we are considering a function f(t; Bt) which depends only on the second variable. Hence there exists an implicit dependence on time since the Brownian motion depends on time. dBt If the differentiation dt existed, then we can easily do this by using chain rule: dB df = t · f 0(B ) dt: dt t We already know that the formula above makes no sense. One possible way to work around this problem is to try to describe the difference df in terms of the difference dBt. In this case, the equation above becomes 0 (1.1) df = f (Bt)dBt: Our new formula at least makes sense, since there is no need to refer to the dBt differentiation dt which does not exist. The only problem is that it does not quite work. Consider the Taylor expansion of f to obtain (∆x)2 (∆x)3 f(x + ∆x) − f(x) = (∆x) · f 0(x) + f 00(x) + f 000(x) + ··· : 2 6 To deduce Equation (1.1) from this formula, we must be able to say that the significant term is the first term (∆x) · f 0(x) and all other terms are of smaller order of magnitude. -
The Ito Integral
Notes on the Itoˆ Calculus Steven P. Lalley May 15, 2012 1 Continuous-Time Processes: Progressive Measurability 1.1 Progressive Measurability Measurability issues are a bit like plumbing – a bit ugly, and most of the time you would prefer that it remains hidden from view, but sometimes it is unavoidable that you actually have to open it up and work on it. In the theory of continuous–time stochastic processes, measurability problems are usually more subtle than in discrete time, primarily because sets of measures 0 can add up to something significant when you put together uncountably many of them. In stochastic integration there is another issue: we must be sure that any stochastic processes Xt(!) that we try to integrate are jointly measurable in (t; !), so as to ensure that the integrals are themselves random variables (that is, measurable in !). Assume that (Ω; F;P ) is a fixed probability space, and that all random variables are F−measurable. A filtration is an increasing family F := fFtgt2J of sub-σ−algebras of F indexed by t 2 J, where J is an interval, usually J = [0; 1). Recall that a stochastic process fYtgt≥0 is said to be adapted to a filtration if for each t ≥ 0 the random variable Yt is Ft−measurable, and that a filtration is admissible for a Wiener process fWtgt≥0 if for each t > 0 the post-t increment process fWt+s − Wtgs≥0 is independent of Ft. Definition 1. A stochastic process fXtgt≥0 is said to be progressively measurable if for every T ≥ 0 it is, when viewed as a function X(t; !) on the product space ([0;T ])×Ω, measurable relative to the product σ−algebra B[0;T ] × FT . -
Immigration Structures Associated with Dawson-Watanabe Superprocesses
View metadata, citation and similar papers at core.ac.uk brought to you COREby provided by Elsevier - Publisher Connector stochastic processes and their applications ELSEVIER StochasticProcesses and their Applications 62 (1996) 73 86 Immigration structures associated with Dawson-Watanabe superprocesses Zeng-Hu Li Department of Mathematics, Beijing Normal University, Be!ring 100875, Peoples Republic of China Received March 1995: revised November 1995 Abstract The immigration structure associated with a measure-valued branching process may be described by a skew convolution semigroup. For the special type of measure-valued branching process, the Dawson Watanabe superprocess, we show that a skew convolution semigroup corresponds uniquely to an infinitely divisible probability measure on the space of entrance laws for the underlying process. An immigration process associated with a Borel right superpro- cess does not always have a right continuous realization, but it can always be obtained by transformation from a Borel right one in an enlarged state space. Keywords: Dawson-Watanabe superprocess; Skew convolution semigroup; Entrance law; Immigration process; Borel right process AMS classifications: Primary 60J80; Secondary 60G57, 60J50, 60J40 I. Introduction Let E be a Lusin topological space, i.e., a homeomorphism of a Borel subset of a compact metric space, with the Borel a-algebra .N(E). Let B(E) denote the set of all bounded ~(E)-measurable functions on E and B(E) + the subspace of B(E) compris- ing non-negative elements. Denote by M(E) the space of finite measures on (E, ~(E)) equipped with the topology of weak convergence. Forf e B(E) and # e M(E), write ~l(f) for ~Efd/~. -
Martingale Theory
CHAPTER 1 Martingale Theory We review basic facts from martingale theory. We start with discrete- time parameter martingales and proceed to explain what modifications are needed in order to extend the results from discrete-time to continuous-time. The Doob-Meyer decomposition theorem for continuous semimartingales is stated but the proof is omitted. At the end of the chapter we discuss the quadratic variation process of a local martingale, a key concept in martin- gale theory based stochastic analysis. 1. Conditional expectation and conditional probability In this section, we review basic properties of conditional expectation. Let (W, F , P) be a probability space and G a s-algebra of measurable events contained in F . Suppose that X 2 L1(W, F , P), an integrable ran- dom variable. There exists a unique random variable Y which have the following two properties: (1) Y 2 L1(W, G , P), i.e., Y is measurable with respect to the s-algebra G and is integrable; (2) for any C 2 G , we have E fX; Cg = E fY; Cg . This random variable Y is called the conditional expectation of X with re- spect to G and is denoted by E fXjG g. The existence and uniqueness of conditional expectation is an easy con- sequence of the Radon-Nikodym theorem in real analysis. Define two mea- sures on (W, G ) by m fCg = E fX; Cg , n fCg = P fCg , C 2 G . It is clear that m is absolutely continuous with respect to n. The conditional expectation E fXjG g is precisely the Radon-Nikodym derivative dm/dn. -
LECTURES 3 and 4: MARTINGALES 1. Introduction in an Arbitrage–Free
LECTURES 3 AND 4: MARTINGALES 1. Introduction In an arbitrage–free market, the share price of any traded asset at time t = 0 is the expected value, under an equilibrium measure, of its discounted price at market termination t = T . We shall see that the discounted share price at any time t ≤ T may also be computed by an expectation; however, this expectation is a conditional expectation, given the information about the market scenario that is revealed up to time t. Thus, the discounted share price process, as a (random) function of time, has the property that its value at any time t is the conditional expectation of the value at a future time given information available at t.A stochastic process with this property is called a martingale. The theory of martingales (initiated by Joseph Doob, following earlier work of Paul Levy´ ) is one of the central themes of modern probability. In discrete-time finance, the importance of martingales stems largely from the Optional Stopping Formula, which (a) places fundamental constraints on (discounted) share price processes, and (b) provides a computational tool of considerable utility. 2. Filtrations of Probability Spaces Filtrations. In a multiperiod market, information about the market scenario is revealed in stages. Some events may be completely determined by the end of the first trading period, others by the end of the second, and others not until the termination of all trading. This suggests the following classification of events: for each t ≤ T , (1) Ft = {all events determined in the first t trading periods}. The finite sequence (Ft)0≤t≤T is a filtration of the space Ω of market scenarios.