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- Title
A Computational Approach to First Passage Problems of Reflected Hyperexponential Jump Diffusion Processes.
- Authors
Cai, Ning; Yang, Xuewei
- Abstract
Statement of Scope and Mission Reflected hyperexponential jump diffusion processes (RHEPs) have broad applications in operations research, financial engineering, insurance, economics, and queueing. Nonetheless, the extant literature lacks efficiently computable formulae for quantities related to first passage problems of RHEPs. This paper provides a new approach to derive such expressions. As a by-product, we obtain a formula for the Laplace transform of the RHEP transition density. Our approach relies on solving specific OIDEs numerically, applying martingale methods, conducting expansions, and performing matrix analysis. Numerical results show that, for cases where rounding error matters, our approach can provide effective approximation that is substantially more efficient than using "multiprecision computing." The extant literature on first passage problems of reflected hyperexponential jump diffusion processes (RHEPs) lacks efficiently computable formulae for the Laplace transform of the joint distribution of the RHEP and its first passage time, cumulative distribution function of the overshoot, expected cumulative value of the discounted increments of the local time up to the first passage time, expected cumulative discounted value of the RHEP up to the first passage time, and expectation of the first passage time. We combine numerical solutions to ordinary integro-differential equations and martingale methods in a novel manner to derive such expressions. For some of these quantities, our approach can deal with the subtle case in which both the RHEP's overall drift and the discount rate equal zero. As a by-product, we obtain a formula for the Laplace transform of the RHEP transition density. We illustrate the numerical performance of our methodology through a few examples. We observe that, when the RHEP's overall drift and the discount rate are very close to zero, rounding errors can make the evaluation of some of our formulae unreliable. In these situations our exact expression for the case in which the RHEP's overall drift and discount rate are both zero can be an effective approximation for the quantities in question that is substantially more efficient than reliably calculating them using their exact expressions and "multiprecision computing." Our research has applications in financial engineering, insurance, economics, and queueing.
- Subjects
DIFFUSION processes; JUMP processes; LAPLACE distribution; NUMERICAL solutions to integro-differential equations; LEVY processes; OPERATIONS research; CUMULATIVE distribution function; FINANCIAL engineering
- Publication
INFORMS Journal on Computing, 2021, Vol 33, Issue 1, p216
- ISSN
1091-9856
- Publication type
Article
- DOI
10.1287/ijoc.2020.0980