Smooth approximation of stochastic differential equations

Abstract

Consider an Ito 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 Wnis chosen. In applications, a natural class of smooth approximations arise by setting Wn(t)=n−1/2∫nt0v∘ϕsds where ϕt is a flow (generated, e.g., by an ordinary differential equation) and v is a mean zero observable. Under mild conditions on ϕt, 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 ϕt, 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 ϕt. The methods used in this paper are a combination of rough path theory and smooth ergodic theory.