Exit problems for jump processes with applications to dividend problems

This paper investigates the first passage times to flat boundaries for hyper-exponential jump (diffusion) processes. Explicit solutions of the Laplace transforms of the distribution of the first passage times, the joint distribution of the first passage times and undershoot (overshoot), the joint distribution of the process and running suprema (infima) are obtained. The processes recover many models appearing in the literature such as the compound Poisson risk models, the diffusion perturbed compound Poisson risk models, and their dual models. As applications, we present explicit expressions of the dividend formulae for barrier strategy and threshold strategy.