Models Beyond the Dirichlet Process
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Introduction to Lévy Processes
Introduction to Lévy Processes Huang Lorick [email protected] Document type These are lecture notes. Typos, errors, and imprecisions are expected. Comments are welcome! This version is available at http://perso.math.univ-toulouse.fr/lhuang/enseignements/ Year of publication 2021 Terms of use This work is licensed under a Creative Commons Attribution 4.0 International license: https://creativecommons.org/licenses/by/4.0/ Contents Contents 1 1 Introduction and Examples 2 1.1 Infinitely divisible distributions . 2 1.2 Examples of infinitely divisible distributions . 2 1.3 The Lévy Khintchine formula . 4 1.4 Digression on Relativity . 6 2 Lévy processes 8 2.1 Definition of a Lévy process . 8 2.2 Examples of Lévy processes . 9 2.3 Exploring the Jumps of a Lévy Process . 11 3 Proof of the Levy Khintchine formula 19 3.1 The Lévy-Itô Decomposition . 19 3.2 Consequences of the Lévy-Itô Decomposition . 21 3.3 Exercises . 23 3.4 Discussion . 23 4 Lévy processes as Markov Processes 24 4.1 Properties of the Semi-group . 24 4.2 The Generator . 26 4.3 Recurrence and Transience . 28 4.4 Fractional Derivatives . 29 5 Elements of Stochastic Calculus with Jumps 31 5.1 Example of Use in Applications . 31 5.2 Stochastic Integration . 32 5.3 Construction of the Stochastic Integral . 33 5.4 Quadratic Variation and Itô Formula with jumps . 34 5.5 Stochastic Differential Equation . 35 Bibliography 38 1 Chapter 1 Introduction and Examples In this introductive chapter, we start by defining the notion of infinitely divisible distributions. We then give examples of such distributions and end this chapter by stating the celebrated Lévy-Khintchine formula. -
Perturbed Bessel Processes Séminaire De Probabilités (Strasbourg), Tome 32 (1998), P
SÉMINAIRE DE PROBABILITÉS (STRASBOURG) R.A. DONEY JONATHAN WARREN MARC YOR Perturbed Bessel processes Séminaire de probabilités (Strasbourg), tome 32 (1998), p. 237-249 <http://www.numdam.org/item?id=SPS_1998__32__237_0> © Springer-Verlag, Berlin Heidelberg New York, 1998, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l’accord avec les conditions générales d’utili- sation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou im- pression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Perturbed Bessel Processes R.A.DONEY, J.WARREN, and M.YOR. There has been some interest in the literature in Brownian Inotion perturbed at its maximum; that is a process (Xt ; t > 0) satisfying (0.1) , where Mf = XS and (Bt; t > 0) is Brownian motion issuing from zero. The parameter a must satisfy a 1. For example arc-sine laws and Ray-Knight theorems have been obtained for this process; see Carmona, Petit and Yor [3], Werner [16], and Doney [7]. Our initial aim was to identify a process which could be considered as the process X conditioned to stay positive. This new process behaves like the Bessel process of dimension three except when at its maximum and we call it a perturbed three-dimensional Bessel process. We establish Ray-Knight theorems for the local times of this process, up to a first passage time and up to infinity (see Theorem 2.3), and observe that these descriptions coincide with those of the local times of two processes that have been considered in Yor [18]. -
Patterns in Random Walks and Brownian Motion
Patterns in Random Walks and Brownian Motion Jim Pitman and Wenpin Tang Abstract We ask if it is possible to find some particular continuous paths of unit length in linear Brownian motion. Beginning with a discrete version of the problem, we derive the asymptotics of the expected waiting time for several interesting patterns. These suggest corresponding results on the existence/non-existence of continuous paths embedded in Brownian motion. With further effort we are able to prove some of these existence and non-existence results by various stochastic analysis arguments. A list of open problems is presented. AMS 2010 Mathematics Subject Classification: 60C05, 60G17, 60J65. 1 Introduction and Main Results We are interested in the question of embedding some continuous-time stochastic processes .Zu;0Ä u Ä 1/ into a Brownian path .BtI t 0/, without time-change or scaling, just by a random translation of origin in spacetime. More precisely, we ask the following: Question 1 Given some distribution of a process Z with continuous paths, does there exist a random time T such that .BTCu BT I 0 Ä u Ä 1/ has the same distribution as .Zu;0Ä u Ä 1/? The question of whether external randomization is allowed to construct such a random time T, is of no importance here. In fact, we can simply ignore Brownian J. Pitman ()•W.Tang Department of Statistics, University of California, 367 Evans Hall, Berkeley, CA 94720-3860, USA e-mail: [email protected]; [email protected] © Springer International Publishing Switzerland 2015 49 C. Donati-Martin et al. -
A Mathematical Theory of Network Interference and Its Applications
INVITED PAPER A Mathematical Theory of Network Interference and Its Applications A unifying framework is developed to characterize the aggregate interference in wireless networks, and several applications are presented. By Moe Z. Win, Fellow IEEE, Pedro C. Pinto, Student Member IEEE, and Lawrence A. Shepp ABSTRACT | In this paper, we introduce a mathematical I. INTRODUCTION framework for the characterization of network interference in In a wireless network composed of many spatially wireless systems. We consider a network in which the scattered nodes, communication is constrained by various interferers are scattered according to a spatial Poisson process impairments such as the wireless propagation effects, and are operating asynchronously in a wireless environment network interference, and thermal noise. The effects subject to path loss, shadowing, and multipath fading. We start introduced by propagation in the wireless channel include by determining the statistical distribution of the aggregate the attenuation of radiated signals with distance (path network interference. We then investigate four applications of loss), the blocking of signals caused by large obstacles the proposed model: 1) interference in cognitive radio net- (shadowing), and the reception of multiple copies of the works; 2) interference in wireless packet networks; 3) spectrum same transmitted signal (multipath fading). The network of the aggregate radio-frequency emission of wireless net- interference is due to accumulation of signals radiated by works; and 4) coexistence between ultrawideband and nar- other transmitters, which undesirably affect receiver nodes rowband systems. Our framework accounts for all the essential in the network. The thermal noise is introduced by the physical parameters that affect network interference, such as receiver electronics and is usually modeled as additive the wireless propagation effects, the transmission technology, white Gaussian noise (AWGN). -
Introduction to Lévy Processes
Introduction to L´evyprocesses Graduate lecture 22 January 2004 Matthias Winkel Departmental lecturer (Institute of Actuaries and Aon lecturer in Statistics) 1. Random walks and continuous-time limits 2. Examples 3. Classification and construction of L´evy processes 4. Examples 5. Poisson point processes and simulation 1 1. Random walks and continuous-time limits 4 Definition 1 Let Yk, k ≥ 1, be i.i.d. Then n X 0 Sn = Yk, n ∈ N, k=1 is called a random walk. -4 0 8 16 Random walks have stationary and independent increments Yk = Sk − Sk−1, k ≥ 1. Stationarity means the Yk have identical distribution. Definition 2 A right-continuous process Xt, t ∈ R+, with stationary independent increments is called L´evy process. 2 Page 1 What are Sn, n ≥ 0, and Xt, t ≥ 0? Stochastic processes; mathematical objects, well-defined, with many nice properties that can be studied. If you don’t like this, think of a model for a stock price evolving with time. There are also many other applications. If you worry about negative values, think of log’s of prices. What does Definition 2 mean? Increments , = 1 , are independent and Xtk − Xtk−1 k , . , n , = 1 for all 0 = . Xtk − Xtk−1 ∼ Xtk−tk−1 k , . , n t0 < . < tn Right-continuity refers to the sample paths (realisations). 3 Can we obtain L´evyprocesses from random walks? What happens e.g. if we let the time unit tend to zero, i.e. take a more and more remote look at our random walk? If we focus at a fixed time, 1 say, and speed up the process so as to make n steps per time unit, we know what happens, the answer is given by the Central Limit Theorem: 2 Theorem 1 (Lindeberg-L´evy) If σ = V ar(Y1) < ∞, then Sn − (Sn) √E → Z ∼ N(0, σ2) in distribution, as n → ∞. -
Fclts for the Quadratic Variation of a CTRW and for Certain Stochastic Integrals
FCLTs for the Quadratic Variation of a CTRW and for certain stochastic integrals No`eliaViles Cuadros (joint work with Enrico Scalas) Universitat de Barcelona Sevilla, 17 de Septiembre 2013 Damped harmonic oscillator subject to a random force The equation of motion is informally given by x¨(t) + γx_(t) + kx(t) = ξ(t); (1) where x(t) is the position of the oscillating particle with unit mass at time t, γ > 0 is the damping coefficient, k > 0 is the spring constant and ξ(t) represents white L´evynoise. I. M. Sokolov, Harmonic oscillator under L´evynoise: Unexpected properties in the phase space. Phys. Rev. E. Stat. Nonlin Soft Matter Phys 83, 041118 (2011). 2 of 27 The formal solution is Z t x(t) = F (t) + G(t − t0)ξ(t0)dt0; (2) −∞ where G(t) is the Green function for the homogeneous equation. The solution for the velocity component can be written as Z t 0 0 0 v(t) = Fv (t) + Gv (t − t )ξ(t )dt ; (3) −∞ d d where Fv (t) = dt F (t) and Gv (t) = dt G(t). 3 of 27 • Replace the white noise with a sequence of instantaneous shots of random amplitude at random times. • They can be expressed in terms of the formal derivative of compound renewal process, a random walk subordinated to a counting process called continuous-time random walk. A continuous time random walk (CTRW) is a pure jump process given by a sum of i.i.d. random jumps fYi gi2N separated by i.i.d. random waiting times (positive random variables) fJi gi2N. -
Levy Processes
LÉVY PROCESSES, STABLE PROCESSES, AND SUBORDINATORS STEVEN P.LALLEY 1. DEFINITIONSAND EXAMPLES d Definition 1.1. A continuous–time process Xt = X(t ) t 0 with values in R (or, more generally, in an abelian topological groupG ) isf called a Lévyg ≥ process if (1) its sample paths are right-continuous and have left limits at every time point t , and (2) it has stationary, independent increments, that is: (a) For all 0 = t0 < t1 < < tk , the increments X(ti ) X(ti 1) are independent. − (b) For all 0 s t the··· random variables X(t ) X−(s ) and X(t s ) X(0) have the same distribution.≤ ≤ − − − The default initial condition is X0 = 0. A subordinator is a real-valued Lévy process with nondecreasing sample paths. A stable process is a real-valued Lévy process Xt t 0 with ≥ initial value X0 = 0 that satisfies the self-similarity property f g 1/α (1.1) Xt =t =D X1 t > 0. 8 The parameter α is called the exponent of the process. Example 1.1. The most fundamental Lévy processes are the Wiener process and the Poisson process. The Poisson process is a subordinator, but is not stable; the Wiener process is stable, with exponent α = 2. Any linear combination of independent Lévy processes is again a Lévy process, so, for instance, if the Wiener process Wt and the Poisson process Nt are independent then Wt Nt is a Lévy process. More important, linear combinations of independent Poisson− processes are Lévy processes: these are special cases of what are called compound Poisson processes: see sec. -
Financial Modeling with L´Evy Processes and Applying L
FINANCIAL MODELING WITH LEVY´ PROCESSES AND APPLYING LEVY´ SUBORDINATOR TO CURRENT STOCK DATA by GONSALGE ALMEIDA Submitted in partial fullfillment of the requirements for the degree of Doctor of Philosophy Dissertation Advisor: Dr. Wojbor A. Woyczynski Department of Mathematics, Applied Mathematics and Statistics CASE WESTERN RESERVE UNIVERSITY January 2020 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of Gonsalge Almeida candidate for the Doctoral of Philosophy degree Committee Chair: Dr.Wojbor Woyczynski Professor, Department of the Mathematics, Applied Mathematics and Statis- tics Committee: Dr.Alethea Barbaro Associate Professor, Department of the Mathematics, Applied Mathematics and Statistics Committee: Dr.Jenny Brynjarsdottir Associate Professor, Department of the Mathematics, Applied Mathematics and Statistics Committee: Dr.Peter Ritchken Professor, Weatherhead School of Management Acceptance date: June 14, 2019 *We also certify that written approval has been obtained for any proprietary material contained therein. CONTENTS List of Figures iv List of Tables ix Introduction . .1 1 Financial Modeling with L´evyProcesses and Infinitely Divisible Distributions 5 1.1 Introduction . .5 1.2 Preliminaries on L´evyprocesses . .6 1.3 Characteristic Functions . .8 1.4 Cumulant Generating Function . .9 1.5 α−Stable Distributions . 10 1.6 Tempered Stable Distribution and Process . 19 1.6.1 Tempered Stable Diffusion and Super-Diffusion . 23 1.7 Numerical Approximation of Stable and Tempered Stable Sample Paths 28 1.8 Monte Carlo Simulation for Tempered α−Stable L´evyprocess . 34 2 Brownian Subordination (Tempered Stable Subordinator) 44 i 2.1 Introduction . 44 2.2 Tempered Anomalous Subdiffusion . 46 2.3 Subordinators . 49 2.4 Time-Changed Brownian Motion . -
Bessel Process, Schramm-Loewner Evolution, and Dyson Model
Bessel process, Schramm-Loewner evolution, and Dyson model – Complex Analysis applied to Stochastic Processes and Statistical Mechanics – ∗ Makoto Katori † 24 March 2011 Abstract Bessel process is defined as the radial part of the Brownian motion (BM) in the D-dimensional space, and is considered as a one-parameter family of one- dimensional diffusion processes indexed by D, BES(D). First we give a brief review of BES(D), in which D is extended to be a continuous positive parameter. It is well-known that D = 2 is the critical dimension such that, when D D (resp. c ≥ c D < Dc), the process is transient (resp. recurrent). Bessel flow is a notion such that we regard BES(D) with a fixed D as a one-parameter family of initial value x> 0. There is another critical dimension Dc = 3/2 and, in the intermediate values of D, Dc <D<Dc, behavior of Bessel flow is highly nontrivial. The dimension D = 3 is special, since in addition to the aspect that BES(3) is a radial part of the three-dimensional BM, it has another aspect as a conditional BM to stay positive. Two topics in probability theory and statistical mechanics, the Schramm-Loewner evolution (SLE) and the Dyson model (i.e., Dyson’s BM model with parameter β = 2), are discussed. The SLE(D) is introduced as a ‘complexification’ of Bessel arXiv:1103.4728v1 [math.PR] 24 Mar 2011 flow on the upper-half complex-plane, which is indexed by D > 1. It is explained (D) (D) that the existence of two critical dimensions Dc and Dc for BES makes SLE have three phases; when D D the SLE(D) path is simple, when D <D<D ≥ c c c it is self-intersecting but not dense, and when 1 < D D it is space-filling. -
A Two-Dimensional Oblique Extension of Bessel Processes Dominique Lepingle
A two-dimensional oblique extension of Bessel processes Dominique Lepingle To cite this version: Dominique Lepingle. A two-dimensional oblique extension of Bessel processes. 2016. hal-01164920v2 HAL Id: hal-01164920 https://hal.archives-ouvertes.fr/hal-01164920v2 Preprint submitted on 22 Nov 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES Dominique L´epingle1 Abstract. We consider a Brownian motion forced to stay in the quadrant by an electro- static oblique repulsion from the sides. We tackle the question of hitting the corner or an edge, and find product-form stationary measures under a certain condition, which is reminiscent of the skew-symmetry condition for a reflected Brownian motion. 1. Introduction In the present paper we study existence and properties of a new process with values in the nonnegative quadrant S = R+ ×R+ where R+ := [0; 1). It may be seen as a two-dimensional extension of a usual Bessel process. It is a two-dimensional Brownian motion forced to stay in the quadrant by electrostatic repulsive forces, in the same way as in the one-dimensional case where a Brownian motion which is prevented from becoming negative by an electrostatic drift becomes a Bessel process. -
Bessel Processes, Stochastic Volatility, and Timer Options
Mathematical Finance, Vol. 26, No. 1 (January 2016), 122–148 BESSEL PROCESSES, STOCHASTIC VOLATILITY, AND TIMER OPTIONS CHENXU LI Guanghua School of Management, Peking University Motivated by analytical valuation of timer options (an important innovation in realized variance-based derivatives), we explore their novel mathematical connection with stochastic volatility and Bessel processes (with constant drift). Under the Heston (1993) stochastic volatility model, we formulate the problem through a first-passage time problem on realized variance, and generalize the standard risk-neutral valuation theory for fixed maturity options to a case involving random maturity. By time change and the general theory of Markov diffusions, we characterize the joint distribution of the first-passage time of the realized variance and the corresponding variance using Bessel processes with drift. Thus, explicit formulas for a useful joint density related to Bessel processes are derived via Laplace transform inversion. Based on these theoretical findings, we obtain a Black–Scholes–Merton-type formula for pricing timer options, and thus extend the analytical tractability of the Heston model. Several issues regarding the numerical implementation are briefly discussed. KEY WORDS: timer options, volatility derivatives, realized variance, stochastic volatility models, Bessel processes. 1. INTRODUCTION Over the past decades, volatility has become one of the central issues in financial mod- eling. Both the historic volatility derived from time series of past market prices and the implied volatility derived from the market price of a traded derivative (in particular, an option) play important roles in derivatives valuation. In addition to index or stock options, a variety of volatility (or variance) derivatives, such as variance swaps, volatility swaps, and options on VIX (the Chicago board of exchange volatility index), are now actively traded in the financial security markets. -
Asymptotics of Query Strategies Over a Sensor Network
Asymptotics of Query Strategies over a Sensor Network Sanjay Shakkottai∗ February 11, 2004 Abstract We consider the problem of a user querying for information over a sensor network, where the user does not have prior knowledge of the location of the information. We consider three information query strategies: (i) a Source-only search, where the source (user) tries to locate the destination by initiating query which propagates as a continuous time random walk (Brownian motion); (ii) a Source and Receiver Driven “Sticky” Search, where both the source and the destination send a query or an advertisement, and these leave a “sticky” trail to aid in locating the destination; and (iii) where the destination information is spatially cached (i.e., repeated over space), and the source tries to locate any one of the caches. After a random interval of time with average t, if the information is not located, the query times-out, and the search is unsuccessful. For a source-only search, we show that the probability that a query is unsuccessful decays as (log(t))−1. When both the source and the destination send queries or advertisements, we show that the probability that a query is unsuccessful decays as t−5/8. Further, faster polynomial decay rates can be achieved by using a finite number of queries or advertisements. Finally, when a spatially periodic cache is employed, we show that the probability that a query is unsuccessful decays no faster than t−1. Thus, we can match the decay rates of the source and the destination driven search with that of a spatial caching strategy by using an appropriate number of queries.