2008/08/03 by Peter K. Friz, Peter Friz, Harald Oberhauser · 48 citations
Economics, Econometrics and Finance · Mathematics · #Applied mathematics #Approximations of π #Brownian motion #Financial Risk and Volatility Modeling #Iterated function #Mathematical analysis #Mathematics #Ode #Path (computing) #Stochastic differential equation #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Term (time) #Type (biology) #math.CA #math.PR #msc:34K20 #msc:60H99
paper · pdf · doi:10.1016/j.jfa.2009.02.010
published in Journal of Functional Analysis 256(10), 3236-3256 (Elsevier BV) · 18 pages
arxiv created 2008/08/03 · openalex publication_date 2009/03/10 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The Wong-Zakai theorem asserts that ODEs driven by "reasonable" (e.g. piecewise linear) approximations of Brownian motion converge to the corresponding Stratonovich stochastic differential equation. With the aid of rough path analysis, we study "non-reasonable" approximations and go beyond a well-known criterion of [Ikeda--Watanabe, North Holland 1989] in the sense that our result applies to perturbations on all levels, exhibiting additional drift terms involving any iterated Lie brackets of the driving vector fields. In particular, this applies to the approximations by McShane ('72) and Sussmann ('91). Our approach is not restricted to Brownian driving signals. At last, these ideas can be used to prove optimality of certain rough path estimates.