Compound Poisson process

Abstract: A new stochastic process, termed the variance gamma process, is proposed as a model for the uncertainty underlying security prices. The unit period distribution is normal conditional on a variance that is distributed as a gamma variate. Its advantages include long tailedness, continuous-time specification, finite moments of all orders, elliptical multivariate unit period distributions, and good empirical fit. The process is pure jump, approximable by a compound Poisson process with high jump frequency and low jump magnitudes. Applications to option pricing show differential effects for options on the money, compared to in or out of the money. Copyright 1990 by the University of Chicago.

Abstract: In many stochastic models, particularly in queueing theory, Poisson arrivals both observe (see) a stochastic process and interact with it. In particular cases and/or under restrictive assumptions it has been shown that the fraction of arrivals that see the process in some state is equal to the fraction of time the process is in that state. In this paper, we present a proof of this result under one basic assumption: the process being observed cannot anticipate the future jumps of the Poisson process.

Abstract: Introduction What in the World Is a Stochastic Process? How to Characterize a Stochastic Process What Do We Do with a Stochastic Process? Discrete-Time Markov Chains: Transient Behavior Definition and Characterization Examples DTMCs in Other Fields Marginal Distributions Occupancy Times Computation of Matrix Powers DTMCs: First Passage Times Definitions Cumulative Distribution Function of T Absorption Probabilities Expectation of T Generating Function and Higher Moments of T DTMCs: Limiting Behavior Exploring the Limiting Behavior by Examples Irreducibility and Periodicity Recurrence and Transience Determining Recurrence and Transience: Infinite DTMCs Limiting Behavior of Irreducible DTMCs Examples: Limiting Behavior of Infinite State-Space Irreducible DTMCs Limiting Behavior of Reducible DTMCs DTMCs with Costs and Rewards Reversibility Poisson Processes Exponential Distributions Poisson Process: Definitions Event Times in a Poisson Process Superposition and Splitting of Poisson Processes Non-Homogenous Poisson Process Compound Poisson Process Continuous-Time Markov Chains Definitions and Sample Path Properties Examples Transient Behavior: Marginal Distribution Transient Behavior: Occupancy Times Computation of P(t): Finite State-Space Computation of P(t): Infinite State-Space First-Passage Times Exploring the Limiting Behavior by Examples Classification of States Limiting Behavior of Irreducible CTMCs Limiting Behavior of Reducible CTMCs CTMCs with Costs and Rewards Phase-Type Distributions Reversibility Queueing Models Introduction Properties of General Queueing Systems Birth and Death Queues Open Queueing Networks Closed Queueing Networks Single Server Queues Retrial Queue Infinite Server Queue Renewal Processes Introduction Properties of N(t) The Renewal Function Renewal-Type Equation Key Renewal Theorem Recurrence Times Delayed Renewal Processes Alternating Renewal Processes Semi-Markov Processes Renewal Processes with Costs/Rewards Regenerative Processes Markov Regenerative Processes Definitions and Examples Markov Renewal Process and Markov Renewal Function Key Renewal Theorem for MRPs Extended Key Renewal Theorem Semi-Markov Processes: Further Results Markov Regenerative Processes Applications to Queues Diffusion Processes Brownian Motion Sample Path Properties of BM Kolmogorov Equations for Standard Brownian Motion First Passage Times Reflected SBM Reflected BM and Limiting Distributions BM and Martingales Cost/Reward Models Stochastic Integration Stochastic Differential Equations Applications to Finance Epilogue Appendix A: Probability of Events Appendix B: Univariate Random Variables Appendix C: Multivariate Random Variables Appendix D: Generating Functions Appendix E: Laplace-Stieltjes Transforms Appendix F: Laplace Transforms Appendix G: Modes of Convergence Appendix H: Results from Analysis Appendix I: Difference and Differential Equations Answers to Selected Problems References Index Exercises appear at the end of each chapter.

Abstract: : A simple and relatively efficient method for simulating one- dimensional and two-dimensional nonhomogeneous Poisson processes is presented. The method is applicable for any rate function and is based on controlled deletion of points in a Poisson process whose rate function dominates the given rate function. In its simplest implementation, the method obviates the need for numerical integration of the rate function, for ordering of points, and for generation of Poisson variates.

Abstract: Let $X_1, \cdots, X_n$ be an arbitrary sequence of dependent Bernoulli random variables with $P(X_i = 1) = 1 - P(X_i = 0) = p_i.$ This paper establishes a general method of obtaining and bounding the error in approximating the distribution of $\sum^n_{i=1} X_i$ by the Poisson distribution. A few approximation theorems are proved under the mixing condition of Ibragimov (1959), (1962). One of them yields, as a special case and with some improvement, an approximation theorem of Le Cam (1960) for the Poisson binomial distribution. The possibility of an asymptotic expansion is also discussed and a refinement in the independent case obtained. The method is similar to that of Charles Stein (1970) in his paper on the normal approximation for dependent random variables.