Exponential change of measure for general piecewise deterministic Markov processes

Abstract

We consider a general piecewise deterministic Markov process (PDMP) $X=\{X_t\}_{t\geqslant~0}$ with a measure-valued generator $\mathcal{A}$, for which the conditional distribution function of the inter-occurrence time is not necessarily absolutely continuous. A general form of the exponential martingales that are associated with $X$ is given by By considering this exponential martingale to be a likelihood-ratio process, we define a new probability measure and show that the process $X$ is still a general PDMP under the new probability measure. We additionally find the new measure-valued generator and its domain. To illustrate our results, we investigate the continuous-time compound binomial model.