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First passage times of a jump diffusion process

Part of:Integral transforms, operational calculusMarkov processes

Published online by Cambridge University Press:  22 February 2016

S. G. Kou and
Hui Wang
S. G. Kou*
Affiliation:
Columbia University
Hui Wang*
Affiliation:
Brown University
*
∗ Postal address: Department of IEOR, Columbia University, New York, NY 10027, USA. Email address: sk75@columbia.edu
∗∗ Postal address: Division of Applied Mathematics, Brown University, Box F, Providence, RI 02912, USA.
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Abstract

This paper studies the first passage times to flat boundaries for a double exponential jump diffusion process, which consists of a continuous part driven by a Brownian motion and a jump part with jump sizes having a double exponential distribution. Explicit solutions of the Laplace transforms, of both the distribution of the first passage times and the joint distribution of the process and its running maxima, are obtained. Because of the overshoot problems associated with general jump diffusion processes, the double exponential jump diffusion process offers a rare case in which analytical solutions for the first passage times are feasible. In addition, it leads to several interesting probabilistic results. Numerical examples are also given. The finance applications include pricing barrier and lookback options.

Keywords

Renewal theorymartingaledifferential equationintegral equationinfinitesimal generatormarked point processLévy processGaver-Stehfest algorithm

MSC classification

Primary:60J75: Jump processes44A10: Laplace transform
Secondary:60J27: Continuous-time Markov processes on discrete state spaces

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003, pp. 504 - 531
Copyright
Copyright © Applied Probability Trust 2003 

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