Smooth approximation of stochastic differential equations

By David Kelly, Ian Melbourne. 2014

101 citations recorded by the source corpus.

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 Wn has smooth sample paths and that Wn converges weakly to W. A central question in stochastic analysis is to understand the limiting behavior of solutions Xn to the ordinary differential equation dXn=a(Xn)dt+b(Xn)dWn. The classical Wong–Zakai theorem gives sufficient conditions under which Xn converges weakly to X provided that the stochastic 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 ∫b(X)dW depends sensitively on how the smooth approximation Wnis chosen. In applications, a natural class of smooth approximations arise by setting Wn(t)=n−1/2∫nt0v∘ϕsds where ϕt is a flow (generated, e.g., by an ordinary differential equation) and v is a mean zero observable. Under mild conditions on ϕt, we give a definitive answer to the interpretation question for the stochastic integral ∫b(X)dW. Our theory applies to Anosov or Axiom A flows ϕt, 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 ϕt. The methods used in this paper are a combination of rough path theory and smooth ergodic theory.

Subfields: Economics and Econometrics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fbb984bbe70b81e1d9a96009557d2c10a18b1bc1

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Citations in the corpus

A Spectral Approach for Quenched Limit Theorems for Random Expanding Dynamical Systems (2017, unknown): “[7,13,23–25,35,42], and another interesting set of related results on limit theorems occur in the setting of homogenisation [22,30,31].”
Asymptotic behavior of random fitzhugh-nagumo systems driven by colored noise (2017, background): “We also remark that the Wong-Zakai approximations can be used to study the solutions and dynamics of stochastic equations, for which the reader is referred to [33, 36, 45, 46, 47, 48, 56, 57, 39] for more details.”
Martingale–coboundary decomposition for families of dynamical systems (2016, methodology): “When the fast dynamics Tǫ is independent of ǫ, the corresponding result without exactness is proved for partially hyperbolic dynamics using standard pairs/martingale problems [24], and for nonuniformly hyperbolic dynamics using rough path theory [33, 34].”
Canonical RDEs and general semimartingales as rough paths (2017, background): “, in homogenization theory [34, 15, 28] and non-standard approximations to Brownian motion [41, 16].”