MaPhySto
The Danish National Research Foundation:
Network in Mathematical Physics and Stochastics

MPS-RR 2004-14
June 2004

Terminal distributions of skipfree Markov additive processes with absorbtion

by: Søren Asmussen

Abstract

Let X be a Markov additive process with continuous paths and a finite background Markov process J so that X evolves as Brownian motion with drift r(i) and variance \sigma^2(i) when J(t) = i. Assuming that J is eventually absorbed at some state a, the density f(x) of Z=X(\zeta) is found where \zeta is the absorbtion time. The form of f(x) is non-smooth at x=0, but the distributions of Z^+ and Z^- are both of phase-type. The derivation involves concepts and results from fluctuation theory such as the Markov processes obtained by sampling J when X is at a relative maximum or minimum. The details are somewhat different for the fluid case where \sigma^2(i)=0 for all i and the Brownian case \sigma^2(i)>0.

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