Invariant Measure of Approximating Method for Stochastic Functional Differential Equation

Abstract

This paper establishes the sufficient criteria for the stability in distribution of numerical solutions for the stochastic functional differential equation (SFDE) with infinite delay. We first check the mean-square boundedness and convergence of numerical solutions from different initial data and prove the existence of the invariant measure of the approximate segment process under suitable conditions. Then, we show that the numerical stationary distribution converges to the stationary distribution of the underlying solution.

Keywords

Stationary distribution / Infinite delay / Stochastic functional differential equation (SFDE) / 34F05 / 34E10 / 37L55 / 70K65

References

[1] Bao J, Shao J, Yuan C. Approximation of invariant measures for regime-switching diffusions. Potential Anal., 2016, 44: 707-727.

[2] Bao J, Shao J, Yuan C. Invariant probability measures for path-dependent random diffusions. Nonlinear Anal., 2023, 228113201

[3] Bao J, Wang F, Yuan C. Asymptotic log-Harnack inequality and applications for stochastic systems of infinite memory. Stochas. Process. Appl., 2019, 129: 4576-4596.

[4] Hale JK, Kato J. Phase space for retarded equations with infinite delay. Funkcial. Ekvac., 1978, 21(1): 11-41. DOI:

[5] Hale JK, Lunel SMV. Introduction to Functional Differential Equations, 1993, Berlin. Springer.

[6] Hino Y, Naito T, Murakami S. Functional Differential Equations with Infinite Delay, 1991, Berlin. Springer.

[7] Hussein KA. Stability in distribution of numerical solution of neutral stochastic functional differential equations with infinite delay. J. Comput. Appl. Math., 2021, 396113625

[8] Kallenberg O. Foundations of Modern Probability, 1997, New York. Springer

[9] Li X, Ma Q, Yang H, Yuan C. The numerical invariant measure of stochastic differetential equations with Markovian switching. SIAM J. Numer. Anal., 2018, 56: 1435-1455.

[10] Li, X., Mao, X., Song, G.: Explicit approximation of invariant measure for stochastic delay differential equations with the nonlinear diffusion term. J. Theor. Probab. 37(2), 1850–1881 (2024)

[11] Li X, Mao X, Yin G. Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence in p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p$$\end{document}th moment and stability. IMA J. Numer. Anal., 2019, 39: 847-892.

[12] Liu W, Mao X. Numerical stationary distribution and its convergence for nonlinear stochastic differential equations. J. Comput. Appl. Math., 2015, 276: 16-29.

[13] Mao X, Yuan C. Stochastic Differetential Equations with Markovian Switching, 2006, London. Imperial College Press.

[14] Mao X, Yuan C, Yin G. Numerical method for stationary distribution of stochastic differential equations with Markovian switching. J. Comput. Appl. Math., 2005, 174: 1-27.

[15] Mei, H., Yin, G., Wu, F.: Properties of stochastic integro-differential equations with infinite delay: regularity, ergodicity, weak sense Fokker-Planck equations. Stochas. Process. Appl. 126(10), 3102–3123 (2016)

[16] Shao J. Inversion measure and Euler-Maruyama’s approximations of state-dependent regime-switching diffusions. SIAM J. Control. Optim., 2018, 56: 3215-3238.

[17] Shao J, Wang L, Wu Q. Ergodicity and stability of hybrid systems with piecewise constant type state-dependent switching. Stochas. Process. Appl., 2023, 161: 1-23.

[18] Shi, B., Wang, Y., Mao, X., Wu, F.: Approximation of invariant measures of a class of backward Euler-Maruyama scheme for stochastic functional differential equations. J. Differ. Equ. 389, 415–456 (2024)

[19] Tan L, Jin W, Suo Y. Stability in distribution of neutral stochastic functional differential equations. Stat. Probab. Lett., 2015, 105: 27-36.

[20] Wang, F., Yuan, C.: Harnack inequalities for functional SDEs with multiplicative noise and applications. Stochas. Process. Appl. 2692–2710 (2011)

[21] Wang Y, Wu F, Yin G, Zhu C. Stochastic functional differential equations with infinite delay under non-Lipschitz coefficients: existence and uniqueness, Markov property, ergodicity, and asymptotic log-Harnack inequality. Stochas. Anal. Appl., 2022, 149: 1-38. DOI:

[22] Wu F, Xu Y. Stochastic Lotka-Volterra population dynamics with infinite delay. SIAM J. Appl. Math., 2009, 70: 641-657.

[23] Wu F, Yin G, Mei H. Stochastic functional differential equations with infinite delay: existence and uniqueness of solutions, solution maps, Markov properties and ergodicity. J. Differ. Equ., 2017, 262: 1226-1252.

[24] Yang H, Li X. Explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons. J. Differ. Equ., 2018, 265: 2921-2967.

[25] Yuan C, Mao X. Stability in distribution of numerical solutions for stochastic differential equations. Stochas. Anal. Appl., 2004, 22: 1133-1150.

[26] Yuan, C., Mao, X.: Stationary distributions of Euler-Maruyama-type stochastic difference equations with Markovian switching and their convergence. J. Differ. Equ. Appl. 11(1), 29–48 (2005)

[27] Zhou S, Hu Y. Numerical approximation for nonlinear stochastic pantograph equations with Markovian switching. Appl. Math. Comput., 2016, 286: 126-138. DOI:

[28] Zhou S, Jin H. Strong convergence of implicit numerical methods for nonlinear stochastic functional differential equations. J. Comput. Appl. Math., 2017, 324: 241-257.

Funding

Innovative Research Group Project of the National Natural Science Foundation of China(12471279)

Rights & permissions

Shanghai University

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.