Strong convergence order for slow-fast McKean-Vlasov stochastic differential equations

被引：45

作者：

Rockner, Michael ^{[1
,2
]}

Sun, Xiaobin ^{[3
,4
]}

Xie, Yingchao ^{[3
,4
]}

机构：

[1] Univ Bielefeld, Fak Math, D-33501 Bielefeld, Germany

[2] Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China

[3] Jiangsu Normal Univ, Sch Math & Stat, Xuzhou 221116, Jiangsu, Peoples R China

[4] Jiangsu Normal Univ, Res Inst Math Sci, Xuzhou 221116, Jiangsu, Peoples R China

来源：

ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES | 2021年 / 57卷 / 01期

关键词：

Averaging principle; McKean-Vlasov stochastic differential equations; Slow-fast; Poisson equation; Strong convergence order; REACTION-DIFFUSION EQUATIONS; DISTRIBUTION DEPENDENT SDES; AVERAGING PRINCIPLE; POISSON EQUATION; SYSTEMS; APPROXIMATION;

D O I：

10.1214/20-AIHP1087

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

In this paper, we consider the averaging principle for a class of McKean-Vlasov stochastic differential equations with slow and fast time-scales. Under some proper assumptions on the coefficients, we first prove that the slow component strongly converges to the solution of the corresponding averaged equation with convergence order 1/3 using the approach of time discretization. Furthermore, under stronger regularity conditions on the coefficients, we use the technique of Poisson equation to improve the order to 1/2, which is the optimal order of strong convergence in general.

引用

页码：547 / 576

页数：30

共 41 条

[1] Multi-timescale systems and fast-slow analysis [J].

Bertram, Richard ;

Rubin, Jonathan E. .

MATHEMATICAL BIOSCIENCES, 2017, 287 :105-121

[2]

Bogoliubov N. N., 1961, ASYMPTOTIC METHODS T

[3] Orders of convergence in the averaging principle for SPDEs: The case of a stochastically forced slow component [J].

Brehier, Charles-Edouard .

STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 2020, 130 (06) :3325-3368

[4] Strong and weak orders in averaging for SPDEs [J].

Brehier, Charles-Edouard .

STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 2012, 122 (07) :2553-2593

[5] MEAN-FIELD STOCHASTIC DIFFERENTIAL EQUATIONS AND ASSOCIATED PDES [J].

Buckdahn, Rainer ;

Li, Juan ;

Peng, Shige ;

Rainer, Catherine .

ANNALS OF PROBABILITY, 2017, 45 (02) :824-878

[6]

Cardaliaguet P., 2012, NOTES MEAN FIELD GAM

[7] AVERAGING PRINCIPLE FOR NONAUTONOMOUS SLOW-FAST SYSTEMS OF STOCHASTIC REACTION-DIFFUSION EQUATIONS: THE ALMOST PERIODIC CASE [J].

Cerrai, Sandra ;

Lunardi, Alessandra .

SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 2017, 49 (04) :2843-2884

[8] AVERAGING PRINCIPLE FOR SYSTEMS OF REACTION-DIFFUSION EQUATIONS WITH POLYNOMIAL NONLINEARITIES PERTURBED BY MULTIPLICATIVE NOISE [J].

Cerrai, Sandra .

SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 2011, 43 (06) :2482-2518

[9] A KHASMINSKII TYPE AVERAGING PRINCIPLE FOR STOCHASTIC REACTION-DIFFUSION EQUATIONS [J].

Cerrai, Sandra .

ANNALS OF APPLIED PROBABILITY, 2009, 19 (03) :899-948

[10] Averaging principle for a class of stochastic reaction-diffusion equations [J].

Cerrai, Sandra ;

Freidlin, Mark .

PROBABILITY THEORY AND RELATED FIELDS, 2009, 144 (1-2) :137-177

← 1 2 3 4 5 →