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.

中文翻译：

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平均原理$$G$$-布朗运动驱动的平均场随机微分方程

 抽象的

在次线性空间 \(\left(\Omega,\mathcal{H},\widehat{\mathbb{E}}\right)\) 中，我们考虑平均场随机微分方程 (\(G\)-MFSDE) ），也称为 \(G\)-McKean-Vlasov 随机微分方程，它们是 SDE，其中系数不仅取决于未知过程的状态，还取决于其定律。我们的意思是，根据 \(\left(\Omega,\mathcal{H},\widehat{\mathbb{E}}\right)\) 上的随机变量 \(X\) 定律，集合 \(\left\ {P_{X}:P\in\mathcal{P}\right\}\)，其中 \(P_{X}\) 是 \(X\) 相对于 \(P\) 和 \( \mathcal{P}\) 是与次线性期望 \(\widehat{\mathbb{E}}\) 相关的概率族。本文利用不动点定理研究\(G\)-MFSDE解的存在唯一性。为此，我们在定律子集和适应的 Lipchitz 和线性增长条件之间引入了一种新型 Kantorovich 度量。此外，我们证明了平均原理的有效性，并得到了收敛定理，其中平均\(G\)-MFSDE的解在均方意义上收敛于标准解。