A Computational Approach to First Passage Problems of Reflected Hyper-Exponential Jump Diffusion Processes

The extant literature on first passage problems of reflected hyper-exponential 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 of both zero RHEP's overall drift and discount rate 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.