Truncated Euler-Maruyama method for hybrid stochastic functional differential equations with infinite time delay

Jingchao Zhou; Henglei Xu; Xuerong Mao

doi:10.1016/j.cam.2025.116773

Truncated Euler-Maruyama method for hybrid stochastic functional differential equations with infinite time delay

Jingchao Zhou, Henglei Xu^*, Xuerong Mao

^*Corresponding author for this work

Mathematics And Statistics

Research output: Contribution to journal › Article › peer-review

Abstract

Li et al. (2023) developed a new theory to approximate the solution of hybrid stochastic functional differential equations (SFDEs) with infinite time delay via the numerical solution of the corresponding hybrid SFDEs with finite time delay. But hybrid SFDEs were required to be globally Lipschitz continuous. In this paper, we will lift this restriction. Under the local Lipschitz condition and the Khasminskii-type condition, numerical solutions of hybrid SFDEs with infinite time delay will be designed by using the truncated Euler–Maruyama method. The strong convergence and convergence rate of the numerical solutions in 𝐿𝑞 (𝑞 ≥ 2) will be obtained. Finally, an example to stochastic functional volatility model is given to demonstrate the effectiveness of our new theory.

Original language	English
Article number	116773
Number of pages	21
Journal	Journal of Computational and Applied Mathematics
Volume	472
Early online date	27 May 2025
DOIs	https://doi.org/10.1016/j.cam.2025.116773
Publication status	E-pub ahead of print - 27 May 2025

Funding

The authors would also like to thank the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the EPSRC (EP/Z003369/1), Chinese Scholarship Council and Strathclyde University (Ph.D. studentship) for their financial support.

Keywords

infinite time delay
hybrid stochastic functional differential equation
truncated Euler-Maruyama method
local Lipschitz condition
strong convergence

Access to Document

10.1016/j.cam.2025.116773Licence: CC BY 4.0

Zhou-etal-JCAM-2025-Truncated-Euler-Maruyama-method-for-hybrid-stochastic-functional-differential-equationsFinal published version, 1.24 MBLicence: CC BY 4.0

Cite this

@article{d0e020fcd2c4473a99d8b125086ec5f0,

title = "Truncated Euler-Maruyama method for hybrid stochastic functional differential equations with infinite time delay",

abstract = "Li et al. (2023) developed a new theory to approximate the solution of hybrid stochastic functional differential equations (SFDEs) with infinite time delay via the numerical solution of the corresponding hybrid SFDEs with finite time delay. But hybrid SFDEs were required to be globally Lipschitz continuous. In this paper, we will lift this restriction. Under the local Lipschitz condition and the Khasminskii-type condition, numerical solutions of hybrid SFDEs with infinite time delay will be designed by using the truncated Euler–Maruyama method. The strong convergence and convergence rate of the numerical solutions in 𝐿𝑞 (𝑞 ≥ 2) will be obtained. Finally, an example to stochastic functional volatility model is given to demonstrate the effectiveness of our new theory.",

keywords = "infinite time delay, hybrid stochastic functional differential equation, truncated Euler-Maruyama method, local Lipschitz condition, strong convergence",

author = "Jingchao Zhou and Henglei Xu and Xuerong Mao",

year = "2025",

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doi = "10.1016/j.cam.2025.116773",

language = "English",

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TY - JOUR

T1 - Truncated Euler-Maruyama method for hybrid stochastic functional differential equations with infinite time delay

AU - Zhou, Jingchao

AU - Xu, Henglei

AU - Mao, Xuerong

PY - 2025/5/27

Y1 - 2025/5/27

N2 - Li et al. (2023) developed a new theory to approximate the solution of hybrid stochastic functional differential equations (SFDEs) with infinite time delay via the numerical solution of the corresponding hybrid SFDEs with finite time delay. But hybrid SFDEs were required to be globally Lipschitz continuous. In this paper, we will lift this restriction. Under the local Lipschitz condition and the Khasminskii-type condition, numerical solutions of hybrid SFDEs with infinite time delay will be designed by using the truncated Euler–Maruyama method. The strong convergence and convergence rate of the numerical solutions in 𝐿𝑞 (𝑞 ≥ 2) will be obtained. Finally, an example to stochastic functional volatility model is given to demonstrate the effectiveness of our new theory.

AB - Li et al. (2023) developed a new theory to approximate the solution of hybrid stochastic functional differential equations (SFDEs) with infinite time delay via the numerical solution of the corresponding hybrid SFDEs with finite time delay. But hybrid SFDEs were required to be globally Lipschitz continuous. In this paper, we will lift this restriction. Under the local Lipschitz condition and the Khasminskii-type condition, numerical solutions of hybrid SFDEs with infinite time delay will be designed by using the truncated Euler–Maruyama method. The strong convergence and convergence rate of the numerical solutions in 𝐿𝑞 (𝑞 ≥ 2) will be obtained. Finally, an example to stochastic functional volatility model is given to demonstrate the effectiveness of our new theory.

KW - infinite time delay

KW - hybrid stochastic functional differential equation

KW - truncated Euler-Maruyama method

KW - local Lipschitz condition

KW - strong convergence

UR - https://www.sciencedirect.com/journal/journal-of-computational-and-applied-mathematics

U2 - 10.1016/j.cam.2025.116773

DO - 10.1016/j.cam.2025.116773

M3 - Article

SN - 0377-0427

VL - 472

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

M1 - 116773

ER -

Truncated Euler-Maruyama method for hybrid stochastic functional differential equations with infinite time delay

Abstract

Funding

Keywords

Access to Document

Other files and links

Fingerprint

DMS-EPSRC: Stabilization using feedback controls, numerical methods for stochastic systems, and systems with mean-field interactions (EPSRC / NSF scheme)

Cite this

Truncated Euler-Maruyama method for hybrid stochastic functional differential equations with infinite time delay

Abstract

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Projects

DMS-EPSRC: Stabilization using feedback controls, numerical methods for stochastic systems, and systems with mean-field interactions (EPSRC / NSF scheme)

Cite this