Smooth approximation of stochastic differential equations

Abstract

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 $W_n$ has smooth sample paths and that $W_n$ converges weakly to $W$. A central question in stochastic analysis is to understand the limiting behavior of solutions $X_n$ to the ordinary differential equation $d{X}_n=a(X_n)dt+b(X_n)d{W}_n$. The classical Wong--Zakai theorem gives sufficient conditions under which $X_n$ converges weakly to $X$ provided that the stochastic integral $\int b(X)dW$ is given the Stratonovich interpretation. The sufficient conditions are automatic in one dimension, but in higher dimensions the correct interpretation of $\int b(X)dW$ depends sensitively on how the smooth approximation $W_n$ is chosen. In applications, a natural class of smooth approximations arise by setting $W_n(t)=n^{-1/2}{\int }_0^{nt}v\circ {\varphi }_{s}ds$ where ${\varphi }_t$ is a flow (generated, e.g., by an ordinary differential equation) and $v$ is a mean zero observable. Under mild conditions on ${\varphi }_t$, we give a definitive answer to the interpretation question for the stochastic integral $\int b(X)dW$. Our theory applies to Anosov or Axiom A flows ${\varphi }_t$, 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 ${\varphi }_t$. The methods used in this paper are a combination of rough path theory and smooth ergodic theory.

Published at http://dx.doi.org/10.1214/14-AOP979 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Rough path recursions and diffusion approximations

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