Non-standard approximations of the Ito-map

Details Non-standard approximations of the Ito-map Non-standard approximations of the Ito-map

2008-08-03

arxiv.org/abs/0808.0337v1

Journal of Functional Analysis 256 (2009), pp. 3236-3256 (different title and minor changes)

Mathematics

Probability

18 pages

Scientific paper

10.1016/j.jfa.2009.02.010

The Wong-Zakai theorem asserts that ODEs driven by "reasonable" (e.g. piecewise linear) approximations of Brownian motion converge to the corresponding Stratonovich stochastic differential equation. With the aid of rough path analysis, we study "non-reasonable" approximations and go beyond a well-known criterion of [Ikeda--Watanabe, North Holland 1989] in the sense that our result applies to perturbations on all levels, exhibiting additional drift terms involving any iterated Lie brackets of the driving vector fields. In particular, this applies to the approximations by McShane ('72) and Sussmann ('91). Our approach is not restricted to Brownian driving signals. At last, these ideas can be used to prove optimality of certain rough path estimates.

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