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Convergence of the tamed-Euler-Maruyama method for SDEs with discontinuous and polynomially growing drift

Kathrin Spendier, Michaela Szölgyenyi

TL;DR

This work extends strong convergence results for the tamed-Euler-Maruyama method to SDEs with drift that is both discontinuous and polynomially growing. It combines a transformation approach with taming to obtain a transformed SDE with improved regularity, and it analyzes both the transformed discretization error and occupation-time crossings near discontinuities. The main contribution is proving strong convergence of order $1/2$ for scalar SDEs with finitely many drift discontinuities, under a set of conditions on the drift and diffusion. This advances numerical analysis for irregular coefficients by unifying two previously separate irregularity regimes and supports stable, provably accurate simulations in applications with piecewise Lipschitz and superlinear drift.

Abstract

Numerical methods for SDEs with irregular coefficients are intensively studied in the literature, with different types of irregularities usually being attacked separately. In this paper we combine two different types of irregularities: polynomially growing drift coefficients and discontinuous drift coefficients. For SDEs that suffer from both irregularities we prove strong convergence of order $1/2$ of the tamed-Euler-Maruyama scheme from [Hutzenthaler, M., Jentzen, A., and Kloeden, P. E., The Annals of Applied Probability, 22(4):1611-1641, 2012].

Convergence of the tamed-Euler-Maruyama method for SDEs with discontinuous and polynomially growing drift

TL;DR

This work extends strong convergence results for the tamed-Euler-Maruyama method to SDEs with drift that is both discontinuous and polynomially growing. It combines a transformation approach with taming to obtain a transformed SDE with improved regularity, and it analyzes both the transformed discretization error and occupation-time crossings near discontinuities. The main contribution is proving strong convergence of order for scalar SDEs with finitely many drift discontinuities, under a set of conditions on the drift and diffusion. This advances numerical analysis for irregular coefficients by unifying two previously separate irregularity regimes and supports stable, provably accurate simulations in applications with piecewise Lipschitz and superlinear drift.

Abstract

Numerical methods for SDEs with irregular coefficients are intensively studied in the literature, with different types of irregularities usually being attacked separately. In this paper we combine two different types of irregularities: polynomially growing drift coefficients and discontinuous drift coefficients. For SDEs that suffer from both irregularities we prove strong convergence of order of the tamed-Euler-Maruyama scheme from [Hutzenthaler, M., Jentzen, A., and Kloeden, P. E., The Annals of Applied Probability, 22(4):1611-1641, 2012].
Paper Structure (7 sections, 13 theorems, 102 equations)

This paper contains 7 sections, 13 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Convergence result
  3. Structure of the proof
  4. Properties of the transformed SDE
  5. Preparatory lemmas
  6. Estimation of the occupation time and the discontinuity crossing probabilities of the tamed-Euler--Maruyama process
  7. Proof of Theorem \ref{['SpSz:mainmain']}

Key Result

Theorem 2.1

Let Assumption 1 hold. Then there exists $C^{(\text{tEM})}\in(0,\infty)$ such that for all $\delta \in(0,1)$ sufficiently small,

Theorems & Definitions (19)

  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • ...and 9 more