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L2 convergence of smooth approximations of Stochastic Differential Equations with unbounded coefficients

Sahani Pathiraja

TL;DR

The paper addresses the problem of obtaining $L^2$ convergence rates for Wong-Zakai type approximations of SDEs with unbounded coefficients. It combines stochastic analysis with rough-path techniques to prove mean-square convergence of nested piecewise linear Brownian approximations to the Stratonovich SDE, under regularity and growth conditions on the drift and diffusion coefficients. The approach uses localisation in state space, rough-path driven RDE analysis, and a Moore–Osgood argument to interchange limits, yielding uniform-in-$d$ moment bounds and the desired $L^2$ convergence for any finite time horizon. The results extend the literature beyond uniformly bounded coefficients and have implications for Monte Carlo and other simulation methods that rely on smooth approximations of Brownian paths, while suggesting applicability to a broader class of piecewise smooth approximations.

Abstract

The aim of this paper is to obtain convergence in mean in the uniform topology of piecewise linear approximations of Stochastic Differential Equations (SDEs) with $C^1$ drift and $C^2$ 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 $L^2$ 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.

L2 convergence of smooth approximations of Stochastic Differential Equations with unbounded coefficients

TL;DR

The paper addresses the problem of obtaining convergence rates for Wong-Zakai type approximations of SDEs with unbounded coefficients. It combines stochastic analysis with rough-path techniques to prove mean-square convergence of nested piecewise linear Brownian approximations to the Stratonovich SDE, under regularity and growth conditions on the drift and diffusion coefficients. The approach uses localisation in state space, rough-path driven RDE analysis, and a Moore–Osgood argument to interchange limits, yielding uniform-in- moment bounds and the desired convergence for any finite time horizon. The results extend the literature beyond uniformly bounded coefficients and have implications for Monte Carlo and other simulation methods that rely on smooth approximations of Brownian paths, while suggesting applicability to a broader class of piecewise smooth approximations.

Abstract

The aim of this paper is to obtain convergence in mean in the uniform topology of piecewise linear approximations of Stochastic Differential Equations (SDEs) with drift and 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 convergence appears yet to be established and is of importance for several applications involving Monte-Carlo approximations. We consider 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.

Paper Structure

This paper contains 5 sections, 7 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of Main Result
  3. Notation and Background on rough path theory
  4. Proof of Theorem \ref{['theo: sdeconv']}
  5. Appendix

Key Result

Theorem 1.1

Assume (ass:bsig) and (ass:correc). Suppose $X_t^d$ is the unique solution of and $X_t$ is the unique strong solution of with $X_0^d = X_0 = x_0$ for some fixed but arbitrary $x_0$. It holds that for every $T > 0$,

Figures (1)

  • Figure 3.1: Limit diagram. Convergence is always $L^2$ with respect to the uniform metric over $[0,T]$

Theorems & Definitions (12)

  • Theorem 1.1
  • Remark 1.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 2 more