First passage process of a Markov additive process, with applications to reflection problems

摘要

In this paper we consider the first passage process of a spectrally negativeMarkov additive process (MAP). The law of this process is uniquelycharacterized by a certain matrix function, which plays a crucial role influctuation theory. We show how to identify this matrix using the theory ofJordan chains associated with analytic matrix functions. Importantly, ourresult also provides us with a technique, which can be used to derive variousfurther identities. We then proceed to show how to compute the stationarydistribution associated with a one-sided reflected (at zero) MAP for both thespectrally positive and spectrally negative cases as well as for the two sidedreflected Markov-modulated Brownian motion; these results can be interpreted interms of queues with MAP input.