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Implicit numerical approximation for stochastic delay differential equations with the nonlinear diffusion term in the infinite horizon

Yudong Wang, Hongjiong Tian

TL;DR

The paper addresses long-horizon numerical approximation of stochastic delay differential equations with nonlinear diffusion under generalized monotonicity and Khasminskii-type conditions. It develops an implicit backward Euler-Maruyama scheme and proves uniform moment bounds and strong convergence at rate $\tfrac{1}{2}$ for $t\ge 0$, matching finite-horizon optimal rates. It further analyzes the segment process, proving uniform convergence in probability of the numerical segment process and that its transition measures converge to the SDDE's invariant measure in the bounded-Lipschitz metric as the step size $\Delta\to 0$. A numerical example corroborates the theory, showing stable long-time behavior, accurate invariant-measure approximation via KS tests, and observed ergodicity, highlighting the method's practicality for long-term SDDE simulations.

Abstract

This paper investigates the approximation of stochastic delay differential equations (SDDEs) via the backward Euler-Maruyama (BEM) method under generalized monotonicity and Khasminskii-type conditions in the infinite horizon. First, by establishing the uniform moment boundedness and finite-time strong convergence of the BEM method, we prove that for sufficiently small step sizes, the numerical approximations strongly converge to the underlying solution in the infinite horizon with a rate of $1/2$, which coincides with the optimal finite-time strong convergence rate. Next, we establish the uniform boundedness and convergence in probability for the segment processes associated with the BEM method. This analysis further demonstrates that the probability measures of the numerical segment processes converge to the underlying invariant measure of the SDDEs. Finally, a numerical example and simulations are provided to illustrate the theoretical results.

Implicit numerical approximation for stochastic delay differential equations with the nonlinear diffusion term in the infinite horizon

TL;DR

The paper addresses long-horizon numerical approximation of stochastic delay differential equations with nonlinear diffusion under generalized monotonicity and Khasminskii-type conditions. It develops an implicit backward Euler-Maruyama scheme and proves uniform moment bounds and strong convergence at rate for , matching finite-horizon optimal rates. It further analyzes the segment process, proving uniform convergence in probability of the numerical segment process and that its transition measures converge to the SDDE's invariant measure in the bounded-Lipschitz metric as the step size . A numerical example corroborates the theory, showing stable long-time behavior, accurate invariant-measure approximation via KS tests, and observed ergodicity, highlighting the method's practicality for long-term SDDE simulations.

Abstract

This paper investigates the approximation of stochastic delay differential equations (SDDEs) via the backward Euler-Maruyama (BEM) method under generalized monotonicity and Khasminskii-type conditions in the infinite horizon. First, by establishing the uniform moment boundedness and finite-time strong convergence of the BEM method, we prove that for sufficiently small step sizes, the numerical approximations strongly converge to the underlying solution in the infinite horizon with a rate of , which coincides with the optimal finite-time strong convergence rate. Next, we establish the uniform boundedness and convergence in probability for the segment processes associated with the BEM method. This analysis further demonstrates that the probability measures of the numerical segment processes converge to the underlying invariant measure of the SDDEs. Finally, a numerical example and simulations are provided to illustrate the theoretical results.
Paper Structure (5 sections, 12 theorems, 75 equations, 2 figures)

This paper contains 5 sections, 12 theorems, 75 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Some properties of the underlying solution
  4. Main results
  5. Numerical examples

Key Result

Lemma 2.4

Under Assumption as2, $\{X_{t_k}\}_{k \geq 0}$ is a homogeneous Markov chain, i.e., for any $B \in \mathfrak{B}(C)$ and $\xi \in \mathcal{C}$ and

Figures (2)

  • Figure 1: The asymptotic behaviour of the error constant $C$ and the K-S statistics for Example \ref{['EX1']};
  • Figure 2: Sample means of $\mathbb{E}(|f(z(k))|)$$(k\in \mathbb{N})$ with different initial data $\xi$

Theorems & Definitions (22)

  • Lemma 2.4
  • Definition 2.5
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 12 more