Exit problems for Kou's process in a Markovian environment

In this paper, we consider a path-wise sum of a Brownian motion plus a compound Poisson process with exponentially distributed positive and negative jumps with parameters that depend on some finite Markov chain. Using known fluctuation identities we investigate one-sided and two-sided exit problems generalizing some results for Kou's processes to the setting of regime switching models without exploiting the fluid embedding technique. The generating function for the hitting time of the state-dependent levels is analyzed. For the case of two states, the numerical examples are given.