A technique for exponential change of measure for Markov processes

Abstract

We consider Markov process fX(t); t 0g with the extended generator A and the domain D(A). Let fF t g be the right-continuous history filtration and by IP t we denote the restriction of IP to fF t g. Let ~ IP be another probability measure on(\\Omega ; F) such that d ~ IP t =dIP t = M (t), where M (t) = h(X(t)) h(X(0)) exp ` \\Gamma Z t 0 (Ah)(X(s)) h(X(s)) ds ' ; is a true martingale for a positive function h 2 D(A). In this note we demonstrate that process fX(t); t 0g is a Markov process on the probability space(\\Omega ; F ; ~ IP) and we show its extended generator ~ A. We apply this result to CTMC, PDMP and to diffusion processes (in this case a special choice of h yields the classical CameronMartin -Girsanov Theorem). Keywords: Markov process, extended generator, exponential change of measure, CameronMartin -Girsanov theorem, (local) martingale. AMS 1991 Subject Classification: Primary: 60G44, 60G25, 60J25, 60J35; Secondary: 60G48, 60H05, 60J57. This work was partiall...