Development of numerical methods for nonlinear hybrid stochastic functional differential equations with infinite delay

2025/08/30 by Guozhen Li, Xiaoyue Li, Li, Guozhen +3
Decision Sciences · Economics, Econometrics and Finance · Physics and Astronomy · #FOS: Mathematics #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Stochastic processes and financial applications #stochastic dynamics and bifurcation

paper · pdf · doi:10.48550/arxiv.2509.00475

openalex publication_date 2025/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper addresses the challenging numerical simulation of nonlinear hybrid stochastic functional differential equations with infinite delays. We first propose an explicit scheme using space and time truncation, requiring only finite historical storage. Leveraging approximation theory, we prove the boundedness of the numerical solution's pth moment and establish its convergence, achieving a rate of 1/2 order under polynomially growing coefficients. Furthermore, we refine the scheme to better capture the underlying exponential stability of the exact solution, in both moment and almost sure senses. Finally, numerical experiments are presented to validate our theoretical results.

Citations

• Hybrid Stochastic Functional Differential Equations with Infinite Delay: Approximations and Numerics
• An explicit approximation for super-linear stochastic functional differential equations
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• Long time behavior of diffusions with Markov switching
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• Fast and Oblivious Convolution Quadrature
• Longtime behaviors of θ-Euler-Maruyama method for stochastic functional differential equations
• Zur Theorie der elastischen Nachwirkung

Related

• Hybrid Stochastic Functional Differential Equations with Infinite Delay: Approximations and Numerics
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• Implicit numerical approximation for stochastic delay differential equations with the nonlinear diffusion term in the infinite horizon
• Existence and uniqueness of solutions to stochastic functional differential equations in infinite dimensions