Some Explicit Results on First Exit Times for a Jump Diffusion Process Involving Semimartingale Local Time

Abstract

In this paper, we consider the one-sided and the two-sided first exit problem for a jump diffusion process with semimartingale local time. Denote this process by $$X=\{X_{t},t\ge 0\}$$ and set $$\tau _{l}=\inf \{t\ge 0, X_{t}\le l\}$$ and $$\tau _{l,u}=\inf \{t\ge 0, X_{t}\notin (l,u)\}$$ with $$l<u$$ . We first establish the existence and uniqueness of strong solutions of the stochastic differential equation (SDE) satisfied by X. Then we investigate the Laplace transforms associated with $$\tau _{l}$$ and $$\tau _{l,u}$$ . It turns out that the explicit expressions for those Laplace transforms can be expressed in terms of exponential functions.