  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.

Probability;Classical Analysis and ODEs