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Strong convergence of the adaptive Milstein method for nonlinear stochastic differential equations with piecewise continuous arguments

Yuhang Zhang, Minghui Song, Jiaqi Zhu

TL;DR

This paper addresses the numerical approximation of stochastic differential equations with piecewise continuous arguments (SDEPCAs) that feature superlinear drift and non globally Lipschitz diffusion. It introduces an explicit adaptive Milstein scheme with variable time steps, governed by a step-size function $\Delta(x)$ and a finite-step condition, and proves strong convergence of order 1 in $\mathcal{L}_p$ for $p\ge 2$, matching the classical Milstein rate under global Lipschitz conditions. The analysis establishes moment bounds, finite step counts, and sharp one-step remainder estimates, ensuring both stability and accuracy without a backstop scheme. Numerical experiments on scalar and multi-dimensional SDEPCAs validate the theoretical convergence and demonstrate superior performance of the adaptive Milstein method over adaptive Euler and, in some cases, over tamed Milstein, highlighting its practical impact for nonlinear stochastic systems with delay-like dependence.

Abstract

In this work, an adaptive time-stepping Milstein method is constructed for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift is one-sided Lipschitz continuous and the diffusion does not impose the commutativity condition. It is widely recognized that explicit Euler or Milstein methods may blow up when the system exhibits superlinear growth, and modifications are needed. Hence we propose an adaptive variant to deal with the case of superlinear growth drift coefficient. To the best of our knowledge, this is the first work to develop a numerical method with variable step sizes for nonlinear SDEPCAs. It is proven that the adaptive Milstein method is strongly convergent in the sense of $L_p, p\ge 2$, and the convergence rate is optimal, which is consistent with the order of the explicit Milstein scheme with globally Lipschitz coefficients. Finally, several numerical experiments are presented to support the theoretical analysis.

Strong convergence of the adaptive Milstein method for nonlinear stochastic differential equations with piecewise continuous arguments

TL;DR

This paper addresses the numerical approximation of stochastic differential equations with piecewise continuous arguments (SDEPCAs) that feature superlinear drift and non globally Lipschitz diffusion. It introduces an explicit adaptive Milstein scheme with variable time steps, governed by a step-size function and a finite-step condition, and proves strong convergence of order 1 in for , matching the classical Milstein rate under global Lipschitz conditions. The analysis establishes moment bounds, finite step counts, and sharp one-step remainder estimates, ensuring both stability and accuracy without a backstop scheme. Numerical experiments on scalar and multi-dimensional SDEPCAs validate the theoretical convergence and demonstrate superior performance of the adaptive Milstein method over adaptive Euler and, in some cases, over tamed Milstein, highlighting its practical impact for nonlinear stochastic systems with delay-like dependence.

Abstract

In this work, an adaptive time-stepping Milstein method is constructed for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift is one-sided Lipschitz continuous and the diffusion does not impose the commutativity condition. It is widely recognized that explicit Euler or Milstein methods may blow up when the system exhibits superlinear growth, and modifications are needed. Hence we propose an adaptive variant to deal with the case of superlinear growth drift coefficient. To the best of our knowledge, this is the first work to develop a numerical method with variable step sizes for nonlinear SDEPCAs. It is proven that the adaptive Milstein method is strongly convergent in the sense of , and the convergence rate is optimal, which is consistent with the order of the explicit Milstein scheme with globally Lipschitz coefficients. Finally, several numerical experiments are presented to support the theoretical analysis.

Paper Structure

This paper contains 5 sections, 8 theorems, 113 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. The Milstein scheme
  4. Strong convergence rate of the adaptive Milstein method
  5. Numerical examples

Key Result

Lemma 4

Under Assumptions Local Lipschitz condition-g condition, there is a unique global solution $x(t)$ to Eq.SDEPCA on $t\ge 0$ with initial data $x(0)=x_0$. Moreover, for any $p>0$, the solution has the property that

Figures (3)

  • Figure 1: Loglog plot of mean square errors against timesteps for Example. \ref{['example_1']}.
  • Figure 2: Loglog plot of errors against step sizes for Example. \ref{['example_2']}.
  • Figure 3: Loglog plot of mean square errors against timesteps for Example \ref{['example_3']}.

Theorems & Definitions (21)

  • Lemma 4
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1: Lord2018Maobook
  • Lemma 7: Conall2019arxiv
  • Lemma 8: Conall2019arxivShiryaev1996
  • Theorem 9
  • proof
  • Lemma 10
  • ...and 11 more