Extending the Wong-Zakai theorem to reversible Markov processes

Bass, Richard F., Hambly, B., and Lyons, Terry. "Extending the Wong-Zakai theorem to reversible Markov processes." Journal of the European Mathematical Society 004.3 (2002): 237-269. <http://eudml.org/doc/277479>.

@article{Bass2002,
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 $p<4$.},
author = {Bass, Richard F., Hambly, B., Lyons, Terry},
journal = {Journal of the European Mathematical Society},
keywords = {reversible Markov processes; rough paths; stochastic area; polygonal approximation; stochastic differential equations; finite $p$-variation; reversible Markov processes; rough paths; stochastic area; polygonal approximation; stochastic differential equations; finite -variation},
language = {eng},
number = {3},
pages = {237-269},
publisher = {European Mathematical Society Publishing House},
title = {Extending the Wong-Zakai theorem to reversible Markov processes},
url = {http://eudml.org/doc/277479},
volume = {004},
year = {2002},
}

TY - JOUR
AU - Bass, Richard F.
AU - Hambly, B.
AU - Lyons, Terry
TI - Extending the Wong-Zakai theorem to reversible Markov processes
JO - Journal of the European Mathematical Society
PY - 2002
PB - European Mathematical Society Publishing House
VL - 004
IS - 3
SP - 237
EP - 269
AB - 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 $p<4$.
LA - eng
KW - reversible Markov processes; rough paths; stochastic area; polygonal approximation; stochastic differential equations; finite $p$-variation; reversible Markov processes; rough paths; stochastic area; polygonal approximation; stochastic differential equations; finite -variation
UR - http://eudml.org/doc/277479
ER -