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

2010/06/15 by D'Auria, Bernardo, Ivanovs, Jevgenijs, Kella, Offer +1
#60K25 #60K37 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1006.2965

Abstract

In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. Importantly, our result also provides us with a technique, which can be used to derive various further identities. We then proceed to show how to compute the stationary distribution associated with a one-sided reflected (at zero) MAP for both the spectrally positive and spectrally negative cases as well as for the two sided reflected Markov-modulated Brownian motion; these results can be interpreted in terms of queues with MAP input.

Related

• Occupation densities in solving exit problems for Markov additive processes and their reflections
• Exit Times for a Discrete Markov Additive Process
• Markov-modulated Brownian motion with two reflecting barriers
• On the Refracted-Reflected Spectrally Negative Lévy Processes
• One-sided Markov additive processes with lattice and non-lattice increments