The Technique of the Exponential Change of Measure for Markov Processes the Technique of the Exponential Change of Measure for Markov Processes

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 ltration and by I P t we denote the restriction of IP to fF t g. Let ~ I P be another probability measure on ((; F) such that d ~ I P t =dI P t = M (t), where M (t) = h(X (t)) h(X (0)) exp ? 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 ((; F; ~ I P) and we show its extended generator ~ A. We apply this result to CTMC, PDMP and to diiusion processes (in this case a special choice of h yields the classical Cameron-Martin-Girsanov Theorem).