SMOOTH APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS

Consider an Itô process X satisfying the stochastic differential equation dX = a(X) dt + b(X) dW where a, b are smooth and W is a multidimensional Brownian motion. Suppose that Wn has smooth sample paths and that Wn converges weakly to W. A central question in stochastic analysis is to understand the limiting behavior of solutions Xn to the ordinary differential equation dXn = a(Xn) dt + b(Xn) dWn. The classical Wong-Zakai theorem gives sufficient conditions under which Xn converges weakly to X provided that the stochastic integral ∫ b(X) dW is given the Stratonovich interpretation. The sufficient conditions are automatic in one dimension, but in higher dimensions the correct interpretation of ∫ b(X) dW depends sensitively on how the smooth approximation Wn is chosen. In applications, a natural class of smooth approximations arise by setting Wn (t) = n-1/2 $\smallint _0^{nt}\upsilon o{\phi _s}ds$ where ɸt is a flow (generated, e.g., by an ordinary differential equation) and υ is a mean zero observable. Under mild conditions on ɸt, we give a definitive answer to the interpretation question for the stochastic integral ∫ b(X) dW. Our theory applies to Anosov or Axiom A flows ɸt, as well as to a large class of nonuniformly hyperbolic flows (including the one defined by the well-known Lorenz equations) and our main results do not require any mixing assumptions on ɸt. The methods used in this paper are a combination of rough path theory and smooth ergodic theory.