Exit problems for reflected Markov-modulated Brownian motion

Abstract

Let (X, J) denote a Markov-modulated Brownian motion (MMBM) and denote its supremum process by S. For some a > 0, let σ(a) denote the time when the reflected process Y := S - X first surpasses the level a. Furthermore, let σ_-(a) denote the last time before σ(a) when X attains its current supremum. In this paper we shall derive the joint distribution of S_σ(a), σ_-(a), and σ(a), where the latter two will be given in terms of their Laplace transforms. We also provide some remarks on scale matrices for MMBMs with strictly positive variation parameters. This extends recent results for spectrally negative Lévy processes to MMBMs. Due to well-known fluid embedding and state-dependent killing techniques, the analysis applies to Markov additive processes with phase-type jumps as well. The result is of interest to applications such as the dividend problem in insurance mathematics and the buffer overflow problem in queueing theory. Examples will be given for the former.