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# Smooth approximation of stochastic differential equations

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### Abstract

Consider an It\^{o} 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 $$dX_n=a(X_n)\,dt+b(X_n)\,dW_n$$. 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 $$W_n(t)=n^{-1/2}\int_0^{nt}v\circ\phi_s\,ds$$ where $$\phi_t$$ is a flow (generated, e.g., by an ordinary differential equation) and $$v$$ is a mean zero observable. Under mild conditions on $$\phi_t$$, 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 $$\phi_t$$, 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 $$\phi_t$$. The methods used in this paper are a combination of rough path theory and smooth ergodic theory.

### Author and article information

###### Journal
2014-03-28
2016-02-09
10.1214/14-AOP979
1403.7281

IMS-AOP-AOP979
Annals of Probability 2016, Vol. 44, No. 1, 479-520
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)
math.DS math.PR
vtex

Differential equations & Dynamical systems, Probability