SMOOTH APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS

• 成果类型: Article
• 署名作者: Kelly, David; Melbourne, Ian
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill; New York University; University of Warwick
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP979
• 发表日期: 2016
• 页码: 479-520
• 关键词: sure invariance-principle; LIMIT-THEOREMS; DYNAMICAL-SYSTEMS; STATISTICAL PROPERTIES; moment inequalities; large deviations; skew-product; CONVERGENCE; INTEGRALS; rates

摘要

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 W-n, has smooth sample paths and that Wn converges wealdy 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 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 f 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) integral(nt)(0) v o 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 integral 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.