Stochastic derivatives and generalized h-transforms of Markov processes

Abstract

Let $R$ be a continuous-time Markov process on the time interval $[0,1]$ with values in some state space $X$. We transform this reference process $R$ into $P:=f(X_0)\exp (-\int_0^1 V_t(X_t) dt) g(X_1)\,R$ where $f,g$ are nonnegative measurable functions on X and V is some measurable function on $[0,1]\times X$. It is easily seen that $P$ is also Markov. The aim of this paper is to identify the Markov generator of $P$ in terms of the Markov generator of $R$ and of the additional ingredients: $f,g$ and $V$ in absence of regularity assumptions on $f,g$ and $V.$ As a first step, we show that the extended generator of a Markov process is essentially its stochastic derivative. Then, we compute the stochastic derivative of $P$ to identify its generator, under a finite entropy condition. The abstract results are illustrated with continuous diffusion processes on $\mathbb{R}^d$ and Metropolis algorithms on a discrete space.