Exponential Change of Measure for General Piecewise Deterministic Markov Processes

We consider a general piecewise deterministic Markov process (PDMP) $X={X_t}_{tgeqslant 0}$ with 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 is presented as $$M^f_t=frac{f(X_t)}{f(X_0)}left[mathrm{Sexp}left(int_{(0,t]}frac{mathrm{d}L(mathcal{A}f)_s}{f(X_{s-})}right)right]^{-1}.$$ Using this exponential martingale as a likelihood ratio process, we define a new probability measure. It is shown that the original process remains a general PDMP under the new probability measure. And we find the new measure-valued generator and its domain.