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Higher order numerical methods for SDEs without globally monotone coefficients

Lei Dai, Xiaojie Wang

TL;DR

This work develops a unified framework of stopped increment-tamed time discretizations for SDEs with non-globally monotone coefficients, introducing stopped Euler/Milstein and an order-1.5 variant. By establishing exponential integrability via Lyapunov-type conditions and leveraging perturbation theory, the authors recover sharp strong convergence rates in settings where global monotonicity fails. Specifically, they prove pathwise uniform convergence of order $1$ for the Milstein method and order $\tfrac{3}{2}$ for the higher-order scheme under appropriate smoothness and growth assumptions, complemented by numerical tests on classical nonlinear stochastic models. The results advance reliable high-order numerical methods for a broad class of SDEs encountered in applications, where traditional schemes may diverge without global monotonicity.

Abstract

In the present work, we delve into further study of numerical approximations of SDEs with non-globally monotone coefficients. We design and analyze a new family of stopped increment-tamed time discretization schemes of Euler, Milstein and order 1.5 type for such SDEs. By formulating a novel unified framework, the proposed methods are shown to possess the exponential integrability properties, which are crucial to recovering convergence rates in the non-global monotone setting. Armed with such exponential integrability properties and by the arguments of perturbation estimates, we successfully identify the optimal strong convergence rates of the aforementioned methods in the non-global monotone setting. Numerical experiments are finally presented to corroborate the theoretical results.

Higher order numerical methods for SDEs without globally monotone coefficients

TL;DR

This work develops a unified framework of stopped increment-tamed time discretizations for SDEs with non-globally monotone coefficients, introducing stopped Euler/Milstein and an order-1.5 variant. By establishing exponential integrability via Lyapunov-type conditions and leveraging perturbation theory, the authors recover sharp strong convergence rates in settings where global monotonicity fails. Specifically, they prove pathwise uniform convergence of order for the Milstein method and order for the higher-order scheme under appropriate smoothness and growth assumptions, complemented by numerical tests on classical nonlinear stochastic models. The results advance reliable high-order numerical methods for a broad class of SDEs encountered in applications, where traditional schemes may diverge without global monotonicity.

Abstract

In the present work, we delve into further study of numerical approximations of SDEs with non-globally monotone coefficients. We design and analyze a new family of stopped increment-tamed time discretization schemes of Euler, Milstein and order 1.5 type for such SDEs. By formulating a novel unified framework, the proposed methods are shown to possess the exponential integrability properties, which are crucial to recovering convergence rates in the non-global monotone setting. Armed with such exponential integrability properties and by the arguments of perturbation estimates, we successfully identify the optimal strong convergence rates of the aforementioned methods in the non-global monotone setting. Numerical experiments are finally presented to corroborate the theoretical results.
Paper Structure (8 sections, 10 theorems, 118 equations, 2 figures)

This paper contains 8 sections, 10 theorems, 118 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. A new family of stopped increment-tamed methods
  4. Exponential integrability properties
  5. Optimal strong convergence rates
  6. Convergence rate of stopped increment-tamed Milstein method
  7. Convergence rate of stopped increment-tamed order 1.5 method
  8. Examples and numerical experiments

Key Result

Lemma 3.1

The equation eq:new_stop_tamed_type_methods can be written as an Itô's process where for the Euclidean orthonormal basis $e_1=(1,...,0),...,e_m=(0,...,1)$ and $s \in [t_k,t_{k+1}]$, and In addition, for $x,u\in \mathbb{R}^d$, and

Figures (2)

  • Figure 1: Convergence rates of stopped increment-tamed Milstein method
  • Figure 2: Convergence rates of stopped increment-tamed order $1.5$ method

Theorems & Definitions (10)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 4.1
  • Theorem 4.2
  • Corollary 4.3
  • Theorem 5.1
  • Lemma 5.2
  • Theorem 5.3