On Mean Field Stochastic Differential Equations Driven by $$G$$ -Brownian Motion with Averaging Principle

IF 0.8 Q2 MATHEMATICS Lobachevskii Journal of Mathematics Pub Date : 2024-07-19 DOI:10.1134/s1995080224600985

A. B. Touati, H. Boutabia, A. Redjil

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Abstract

In a sublinear space $\left(\Omega,\mathcal{H},\widehat{\mathbb{E}}\right)$, we consider Mean Field stochastic differential equations ($G$-MFSDEs in short), called also $G$-McKean–Vlasov stochastic differential equations, which are SDEs where coefficients depend not only on the state of the unknown process but also on its law. We mean by law of a random variable $X$ on $\left(\Omega,\mathcal{H},\widehat{\mathbb{E}}\right)$, the set $\left\{P_{X}:P\in\mathcal{P}\right\}$, where $P_{X}$ is the law of $X$ with respect to $P$ and $\mathcal{P}$ is the family of probabilities associated to the sublinear expectation $\widehat{\mathbb{E}}$. In this paper, we study the existence and uniqueness of the solution of $G$-MFSDE by using the fixed point theorem. To this end, we introduce a new type Kantorovich metric between subsets of laws and adapted Lipchitz and linear growth conditions. Furthermore, we prove the validity of the averaging principle and obtain convergence theorem where the solution of the averaged $G$-MFSDE converges to that of the standard one in the mean square sense.

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带平均原则的$$G$$布朗运动驱动的平均场随机微分方程

Abstract In a sublinear space $\left(\Omega,\mathcal{H},\widehat\mathbb{E}}right)$,we consider Mean Field stochastic differential equations (简称$G$-MFSDEs), called also $G$-McKean-Vlasov stochastic differential equations, which are SDEs where cofficients depend on not only the state of unknown process but also on its law.我们所说的随机变量（X）的规律是指（left(\Omega,\mathcal{H},\widehat{mathbb{E}}\right)）上的集合（\left\{P_{X}：其中，$P_{X}$是$X)关于\(P$的规律，而$\mathcal{P}$是与亚线性期望$\widehat\{mathbb{E}}$相关的概率族。本文利用定点定理研究了 $G$-MFSDE 解的存在性和唯一性。为此，我们在定律子集之间引入了一种新型康托洛维奇度量，并调整了李普希兹条件和线性增长条件。此外，我们证明了平均原理的有效性，并得到了收敛定理，即平均 $G$-MFSDE 的解在均方意义上收敛于标准解。

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来源期刊

Lobachevskii Journal of Mathematics MATHEMATICS-

CiteScore

1.50

自引率

42.90%

发文量

127

期刊介绍： Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.

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