A Hyper-Erlang Jump-Diffusion Process and Applications in Finance

Abstract

This paper studies the first passage time problem for a reflected two-sided jump-diffusion risk model with the jumps having a hyper-Erlang distribution. The authors give the explicit closed-form expression for the joint Laplace transform of the first passage time and the overshoot for the reflected process. Finally, the formula is applied to the ruin problem under the barrier dividend strategy and the pricing of the Russian option.

Exit Problems for Jump-Diffusion Processes with Uniform Jumps

“…Jump-diffusion processes are used in mathematical finance as models for the evolution of stock prices (see Merton [9] and Dong and Han [4]), and also in reliability theory (Ghamlouch et al [5]). First-passage-time problems for these jumpdiffusion processes have been considered, among others, by Abundo [1] and Kou and Wang [7] (see also the references therein).…”

“…In Abundo [1], the author assumed that the jumps (positive and/or negative) were of constant size. Kou and Wang [7] proposed an asymmetric double exponential distribution for the jump sizes, while Dong and Han [4] used a hyper-Erlang distribution. Other possibilities are log-normal (Merton [9]) and log-uniform (Ahlip and Prodan [2]) distributions.…”