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January 2016 Smooth approximation of stochastic differential equations
David Kelly, Ian Melbourne
Ann. Probab. 44(1): 479-520 (January 2016). DOI: 10.1214/14-AOP979


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.


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David Kelly. Ian Melbourne. "Smooth approximation of stochastic differential equations." Ann. Probab. 44 (1) 479 - 520, January 2016.


Received: 1 March 2014; Revised: 1 October 2014; Published: January 2016
First available in Project Euclid: 2 February 2016

zbMATH: 1372.60082
MathSciNet: MR3456344
Digital Object Identifier: 10.1214/14-AOP979

Primary: 60H10
Secondary: 37A50, 37D20, 37D25

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 1 • January 2016
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