The Annals of Probability

Smooth approximation of stochastic differential equations

David Kelly and Ian Melbourne

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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 $dX_{n}=a(X_{n})\,dt+b(X_{n})\,dW_{n}$.

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.

Article information

Ann. Probab., Volume 44, Number 1 (2016), 479-520.

Received: March 2014
Revised: October 2014
First available in Project Euclid: 2 February 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]

Interpretation of stochastic integrals Wong–Zakai approximation uniform and nonuniform hyperbolicity rough paths iterated invariance principle


Kelly, David; Melbourne, Ian. Smooth approximation of stochastic differential equations. Ann. Probab. 44 (2016), no. 1, 479--520. doi:10.1214/14-AOP979.

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