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Strong convergence in the infinite horizon of numerical methods for stochastic delay differential equations

Yudong Wang, Hongjiong Tian

TL;DR

This paper develops a general framework for proving strong convergence of numerical methods for stochastic delay differential equations (SDDEs) on the infinite horizon, with a focus on the associated segment processes and invariant measures. By formulating four preconditions (Condition con1 and its variant con19) and applying a horizon-partitioning technique, the authors show that both truncated Euler-Maruyama (TEM) and backward Euler-Maruyama (BEM) schemes achieve time-uniform convergence with rates tied to the step size, and that the numerical segment-process measures converge to the true invariant measures in the Fortet-Mourier distance as the step size vanishes. The results extend to the convergence of invariant measures without requiring separate proofs of numerical-invariant-measure existence, and are supported by numerical experiments illustrating time-uniform error bounds and empirical convergence of distributions. Overall, the work provides a practical route for reliable long-time simulation of SDDEs and their invariant properties using explicit and implicit one-step schemes.

Abstract

In this work, we present a general technique for establishing the strong convergence of numerical methods for stochastic delay differential equations (SDDEs) in the infinite horizon. This technique can also be extended to analyze certain continuous function-valued segment processes associated with the numerical methods, facilitating the numerical approximation of invariant measures of SDDEs. To illustrate the application of these results, we specifically investigate the backward and truncated Euler-Maruyama methods. Several numerical experiments are provided to demonstrate the theoretical results.

Strong convergence in the infinite horizon of numerical methods for stochastic delay differential equations

TL;DR

This paper develops a general framework for proving strong convergence of numerical methods for stochastic delay differential equations (SDDEs) on the infinite horizon, with a focus on the associated segment processes and invariant measures. By formulating four preconditions (Condition con1 and its variant con19) and applying a horizon-partitioning technique, the authors show that both truncated Euler-Maruyama (TEM) and backward Euler-Maruyama (BEM) schemes achieve time-uniform convergence with rates tied to the step size, and that the numerical segment-process measures converge to the true invariant measures in the Fortet-Mourier distance as the step size vanishes. The results extend to the convergence of invariant measures without requiring separate proofs of numerical-invariant-measure existence, and are supported by numerical experiments illustrating time-uniform error bounds and empirical convergence of distributions. Overall, the work provides a practical route for reliable long-time simulation of SDDEs and their invariant properties using explicit and implicit one-step schemes.

Abstract

In this work, we present a general technique for establishing the strong convergence of numerical methods for stochastic delay differential equations (SDDEs) in the infinite horizon. This technique can also be extended to analyze certain continuous function-valued segment processes associated with the numerical methods, facilitating the numerical approximation of invariant measures of SDDEs. To illustrate the application of these results, we specifically investigate the backward and truncated Euler-Maruyama methods. Several numerical experiments are provided to demonstrate the theoretical results.

Paper Structure

This paper contains 12 sections, 18 theorems, 147 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Main theorems
  4. Truncated Euler-Maruyama method
  5. The second condition in Condition \ref{['con1']}
  6. The third and fourth conditions in Condition \ref{['con1']}
  7. Backward Euler-Maruyama method
  8. The second and third conditions in Condition \ref{['con19']}
  9. The fourth condition in Condition \ref{['con19']}
  10. Numerical examples
  11. Proof of Theorem \ref{['T--2']}
  12. Proof of Lemma \ref{['L13']}

Key Result

Theorem 3.3

Suppose Condition con1 holds, then the numerical solution $\{X(t; 0, \xi)\}_{t \geq -\tau}$ converges strongly to the underlying solution of SDDE in the infinite horizon, i.e., for any $t \geq -\tau$ where $C$ is a positive constant independent of T.

Figures (3)

  • Figure 1: Left: The BEM method; Right:The TEM method;
  • Figure 2: Left:Difference between empirical distribution; Right:Empirical density function;
  • Figure 3: Sample means of $\mathbb{E}(|f(z(k))|)$$(k\in \mathbb{N})$ with different initial data $\xi$

Theorems & Definitions (33)

  • Definition 2.1
  • Remark 3.2
  • Theorem 3.3
  • Lemma 3.5
  • Theorem 3.6
  • Theorem 3.7
  • proof
  • Lemma 4.5
  • Lemma 4.6
  • proof
  • ...and 23 more