JournalsjemsVol. 4 , No. 3DOI 10.1007/s100970200040

Extending the Wong-Zakai theorem to reversible Markov processes

  • Richard F. Bass

    University of Connecticut, Storrs, United States
  • Ben Hambly

    Oxford University, United Kingdom
  • Terry J. Lyons

    Oxford University, United Kingdom
Extending the Wong-Zakai theorem to reversible Markov processes cover

Abstract

We show how to construct a canonical choice of stochastic area for paths of reversible Markov processes satisfying a weak Hölder condition, and hence demonstrate that the sample paths of such processes are rough paths in the sense of Lyons. We further prove that certain polygonal approximations to these paths and their areas converge in p-variation norm. As a corollary of this result and standard properties of rough paths, we are able to provide a significant generalization of the classical result of Wong-Zakai on the approximation of solutions to stochastic differential equations. Our results allow us to construct solutions to differential equations driven by reversible Markov processes of finite p-variation with p<4.