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Hybrid Stochastic Functional Differential Equations with Infinite Delay: Approximations and Numerics

Guozhen Li, Xiaoyue Li, Xuerong Mao, Guoting Song

TL;DR

This work establishes a rigorous bridge between hybrid SFDEs with infinite delays and their finite-delay counterparts by truncation. It proves global existence and bounded moments for the infinite-delay system, derives asymptotic and exponential convergence results for the truncated finite-delay models under various structural conditions, and develops EM-type numerical methods for the truncated systems. The results enable accurate numerical approximation of infinite-delay SFDEs, with explicit convergence rates for global-Lipschitz, Khasminskii, and highly nonlinear classes, and are complemented by numerical demonstrations on representative models. Overall, the paper provides a comprehensive theoretical and computational framework for solving hybrid SFDEs with infinite memory.

Abstract

This paper is to investigate if the solution of a hybrid stochastic functional differential equation (SFDE) with infinite delay can be approximated by the solution of the corresponding hybrid SFDE with finite delay. A positive result is established for a large class of highly nonlinear hybrid SFDEs with infinite delay. Our new theory makes it possible to numerically approximate the solution of the hybrid SFDE with infinite delay, via the numerical solution of the corresponding hybrid SFDE with finite delay.

Hybrid Stochastic Functional Differential Equations with Infinite Delay: Approximations and Numerics

TL;DR

This work establishes a rigorous bridge between hybrid SFDEs with infinite delays and their finite-delay counterparts by truncation. It proves global existence and bounded moments for the infinite-delay system, derives asymptotic and exponential convergence results for the truncated finite-delay models under various structural conditions, and develops EM-type numerical methods for the truncated systems. The results enable accurate numerical approximation of infinite-delay SFDEs, with explicit convergence rates for global-Lipschitz, Khasminskii, and highly nonlinear classes, and are complemented by numerical demonstrations on representative models. Overall, the paper provides a comprehensive theoretical and computational framework for solving hybrid SFDEs with infinite memory.

Abstract

This paper is to investigate if the solution of a hybrid stochastic functional differential equation (SFDE) with infinite delay can be approximated by the solution of the corresponding hybrid SFDE with finite delay. A positive result is established for a large class of highly nonlinear hybrid SFDEs with infinite delay. Our new theory makes it possible to numerically approximate the solution of the hybrid SFDE with infinite delay, via the numerical solution of the corresponding hybrid SFDE with finite delay.

Paper Structure

This paper contains 13 sections, 10 theorems, 163 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximations by SFDEs with finite delay
  4. Truncated SFDEs
  5. Asymptotic approximations
  6. Approximations with exponential convergence order
  7. Important classes of SFDEs
  8. Global Lipschitz
  9. Khasminiskii case
  10. Highly nonlinear SFDEs
  11. Numerical methods
  12. Lipschitz case
  13. Highly nonlinear case

Key Result

Theorem 2.3

Under Assumptions A2.1 and A2.2, the SFDE (sfde) with the initial data (id) has a unique solution $x(t)$ on $t\in (-\infty,\infty)$, which has the property that where $C_T$ is a positive constant dependent on $T$.

Figures (2)

  • Figure 1: (a) The mean square error for 1000 sample points between $x(10)$ and $x^k(10)$ as the function of $k\in\{10,11,12,13,14\}$. (b) The root mean square errors for 1000 sample points between $x(10)$ and $X^{50}_\Delta(10)$ as the function of $\Delta \in\{2^{-6}, 2^{-8}, 2^{-10}, 2^{-12}, 2^{-14}\}$.
  • Figure 2: (a) The mean square error for 1000 sample points between $x(10)$ and $x^k(10)$ as the function of $k\in\{10,12,14,16,18\}$. (b) The root mean square error for 1000 sample points between $x(10)$ and $X_\Delta^{30}(10)$ as the function of $\Delta\in\{2^{-6}, 2^{-8}, 2^{-10}, 2^{-12}, 2^{-14}\}$.

Theorems & Definitions (18)

  • Theorem 2.3
  • Theorem 3.1
  • proof
  • Theorem 3.4
  • Theorem 3.6
  • Remark 3.7
  • Lemma 5.4
  • proof
  • Remark 5.5
  • Lemma 5.6
  • ...and 8 more