Exit times for a class of piecewise exponential Markov processes with two-sided jumps

被引:30
作者
Jacobsen, Martin
Jensen, Anders Tolver
机构
[1] Royal Vet & Agr Univ, Dept Nat Sci, DK-1871 Frederiksberg C, Denmark
[2] Univ Copenhagen, Dept Appl Math & Stat, DK-2100 Copenhagen O, Denmark
关键词
shot-noise process; Ornstein-Uhlenbeck process; exit probabilities; undershoot; martingales; partial eigenfunctions; contour integrals;
D O I
10.1016/j.spa.2007.01.005
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
We consider first passage times for piecewise exponential Markov processes that may be viewed as Ornstein-Uhlenbeck processes driven by compound Poisson processes. We allow for two-sided jumps and as a main result we derive the joint Laplace transform of the first passage time of a lower level and the resulting undershoot when passage happens as a consequence of a downward (negative) jump. The Laplace transform is determined using complex contour integrals and we illustrate how the choice of contours depends in a crucial manner on the particular form of the negative jump part, which is allowed to belong to a dense class of probabilities. We give extensions of the main result to two-sided exit problems where the negative jumps are as before but now it is also required that the positive jumps have a distribution of the same type. Further, extensions are given for the case where the driving Levy process is the sum of a compound Poisson process and an independent Brownian motion. Examples are used to illustrate the theoretical results and include the numerical evaluation of some concrete exit probabilities. Also, some of the examples show that for specific values of the model parameters it is possible to obtain closed form expressions for the Laplace transform, as is the case when residue calculus may be used for evaluating the relevant contour integrals. (C) 2007 Elsevier B.V. All rights reserved.
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页码:1330 / 1356
页数:27
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