L 2 convergence of smooth approximations of stochastic differential equations with unbounded coefficients

Abstract

AbstractThe aim of this article is to obtain convergence in mean in the uniform topology of piecewise linear approximations of stochastic differential equations (SDEs) with C1 drift and C2 diffusion coefficients with uniformly bounded derivatives. Convergence analyses for such Wong-Zakai approximations most often assume that the coefficients of the SDE are uniformly bounded. Almost sure convergence in the unbounded case can be obtained using now standard rough path techniques, although Lq convergence appears yet to be established and is of importance for several applications involving Monte-Carlo approximations. We consider L2 convergence in the unbounded case using a combination of traditional stochastic analysis and rough path techniques. We expect our proof technique extend to more general piecewise smooth approximations.Keywords: Wong-Zakaiunbounded coefficientspiecewise smooth approximationsstochastic differential equationsrough paths AcknowledgmentsThe author is grateful to Wilhelm Stannat and Sebastian Reich for helpful feedback on this work, as well as anonymous reviewer for their insightful suggestions which has improved this manuscript.Disclosure statementNo potential conflict of interest was reported by the authors.Notes1. This holds for more general semimartingale drivers, but for the purposes of this article, we focus on the case of Wiener processes.2. the notation pi is used to refer to the ith index of a permutation of the set {1,2,⋯r}3. Here we are working with equivalence classesAdditional informationFundingThis research has been partially funded by Deutsche Forschungsgemeinschaft (DFG)- SFB1294/1 - 318763901.