L² convergence of smooth approximations of stochastic differential equations with unbounded coefficients

Abstract

The 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 L^q 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-Zakai
• unbounded coefficients
• piecewise smooth approximations
• stochastic differential equations
• rough paths

Acknowledgments

The 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 statement

No potential conflict of interest was reported by the authors.

Notes

1. 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 p_i is used to refer to the ith index of a permutation of the set $\{1,2,\cdots r\}$

3. Here we are working with equivalence classes

Additional information

Funding

This research has been partially funded by Deutsche Forschungsgemeinschaft (DFG)- SFB1294/1 - 318763901.