A technique for exponential change of measure for Markov processes

Abstract

We consider a Markov process $X(t)$ with extended generator ${\mathbf{A}}$ and domain $\cal{D}({\mathbf{A}})$. Let $\{{\cal F}_t\}$ be a right-continuous history filtration and ${\mathbb{P}}_t$ denote the restriction of ${\mathbb{P}}$ to ${\cal F}_t$. Let $\tilde{{\mathbb{P}}}$ be another probability measure on $(\Omega,{\cal F})$ such that $\rm d\tilde{{\mathbb{P}}}_t/\rm d{\mathbb{P}}_t=E^h(t)$, where $$ E^h(t)=\frac{h(X(t))}{h(X(0))}\exp \left(-\int_0^t \frac{({\mathbf{A}}h)(X(s))}{h(X(s))}\rm ds \right) $$ is a true martingale for a positive function $h\in\cal{D}({\mathbf{A}})$. We demonstrate that the process $X(t)$ is a Markov process on the probability space $ (\Omega, {\cal F}, \{{\cal F}_t\},\tilde{{\mathbb{P}}}) $, we find its extended generator $\tilde{{\mathbf{A}}}$ and provide sufficient conditions under which $\cal{D}(\tilde{{\mathbf{A}}})=\cal{D}({\mathbf{A}})$. We apply this result to continuous-time Markov chains, to piecewise deterministic Markov processes and to diffusion processes (in this case a special choice of $h$ yields the classical Cameron--Martin--Girsanov theorem).