Representation of Semimartingale Markov Processes in Terms of Wiener Processes and Poisson Random Measures

Abstract

Our object is the representation of Markov processes taking values in IRm in terms of well-understood processes and operations. The major result is that every semimartingale Hunt process is obtained by a random time change from a Markov process that satisfies a stochastic integral equation driven by a Wiener process and a Poisson random measure. If the stochastic equation has no other solutions, then the probability law of the process is specified by four deterministic functions. In the particular case of Hunt processes whose paths are of bounded variation over finite intervals, the representation involves only a Poisson random measure and no stochastic integrals. A further corollary is that every continuous strong Markov process on IRm. whose paths are of bounded variation over finite intervals is totally deterministic except in the choice of initial state.