A technique for exponential change of measure for Markov processes
This paper presents a detailed account of the change of probability measure technique for càdlàg Markov processes. Consider a Markov process \(X(t)\) on a filtered probability space \((\Omega, {\mathcal F}, \{ {\mathcal F}_t \}, P)\) having extended generator \({\mathbf A}\) with domain \({\mathcal D}({\mathbf A}).\) Let \(\tilde{P}\) be a probability measure on \((\Omega, {\mathcal F})\) such that the procees \(d\tilde{P}_t/dP_t = E^h (t)\) is an exponential martingale, \(h\) is a positive function from \({\mathcal D}({\mathbf A}).\) It is shown that the process \(X(t)\) is a Markov process on the probability space \((\Omega, {\mathcal F}, \{ {\mathcal F}_t \}, \tilde{P}).\) Its extended generator \(\tilde{\mathbf A}\) and sufficient conditions under which \({\mathcal D}(\tilde{\mathbf A}) = {\mathcal D}({\mathbf A})\) are found. The authors also accomodate some non-Markovian processes that are Markovian with a supplementary component, for example, piecewise deterministic Markov processes or Markov additive processes. For diffusion processes, a special case of the theory presented is a Girsanov-type theorem.
scientific article; zbMATH DE number 1877110