Two-Sided First Exit Problem for Jump Diffusion Distribution Processes Having Jumps with a Mixture of Erlang

Abstract

In this paper, we consider the two-sided first exit problem for jump diffusion processes having jumps with rational Laplace transforms. We investigate the probabilistic property of conditional memorylessness, and drive the joint distribution of the first exit time from an interval and the overshoot over the boundary at the exit time.

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

“…From Lemma 2.1 in [11], we have the following result. Lemma 2.1 For δ > 0, Equation (3) has exactly N + 1 roots r 1 , r 2 , · · · , r N +1 on the lefthalf complex plane and exactly M + 1 roots ρ 1 , ρ 2 , · · · , ρ M+1 on the right-half complex plane.…”

“…This completes the proof. Remark 3.1 Let A be the coefficient matrix of the linear system (11). If the matrix A is invertible, then c i s, d j s uniquely solve the system (11).…”

“…Remark 3.1 Let A be the coefficient matrix of the linear system (11). If the matrix A is invertible, then c i s, d j s uniquely solve the system (11). However, for the proof of the nonsingularity of A, we have not solved it until now and we will study this problem in the future's research.…”

“…Based on the results, they also presented explicit expressions of the dividend formulae for the barrier strategy and the threshold strategy. Furthermore, Wen and Yin [11] investigated the two-sided first exit problem under the hyper-Erlang jump-diffusion model. They derived the joint distribution of the first exit time from an interval and the overshoot over the boundary at the exit time.…”