Funded by The Danish National Research Foundation

MPS-RR 2004-14

June 2004

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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