CS46: Asymptotic behaviors for McKean-Vlasov Stochastic Differential Equations

Organizer: Huijie Qiao (Southeast University)

Averaging principles and central limit theorems for multiscale McKean-Vlasov stochastic systems

Huijie Qiao

In this paper, we study a class of multiscale McKean-Vlasov stochastic systems where the entire system depends on the distribution of the fast component. First of all, by the Poisson equation method we prove that the slow component converges to the solution of the averaging equation in the \(L^p\) (\(p\geq 2\)) space with the optimal convergence rate \(\frac12\). Then a central limit theorem is established by tightness.

Asymptotic behaviors for Volterra type McKean-Vlasov integral equations with small noise

Shanqi Liu

This work is devoted to studying asymptotic behaviors for Volterra type McKean-Vlasov stochastic differential equations with small noise: \[\begin{aligned} \label{Volterra MV eq} X^\varepsilon_{t,\xi}=\xi+\int_{0}^{t}K_1(t,s)b(s,X^\varepsilon_{s,\xi},\mathcal{L}_{X^\varepsilon_{s,\xi}})\d s+\sqrt{\varepsilon}\int_{0}^{t}K_2\left(t,s\right)\sigma(s,X^\varepsilon_{s,\xi},\mathcal{L}_{X^\varepsilon_{s,\xi}})\d W_s,\quad 0\le t\le T\,, \end{aligned}\] where \(\mathcal{L}_{X^{\varepsilon}_{t,\xi}} \in \mathcal{P}_2(\mathbb{R}^d)\) denotes the probability law of \(X^{\varepsilon}_{t,\xi}\), \(\varepsilon\in(0, 1]\) is a small parameter, \(K_i(t,s)\), \(i = 1, 2\) are two positive functions defined on the simplex \(\Delta_T=\left\{0\le s<t\le T\right\}\) which may be singular, \(W_t\) is an \(\mathrm{m}\)-dimensional Brownian motion defined on the classical Wiener space \((\Omega, \mathcal{F}, \mathbb{P})\), \(\xi\) is a \(\mathbb{R}^d\) valued random variable independent of the Brownian motion \(B\), and the coefficients \(\sigma:[0,T]\times\mathbb{R}^d\times \mathcal{P}_2(\mathbb{R}^d)\to\mathbb{R}^d\times \mathbb{R}^m\), \(b:[0,T]\times\mathbb{R}^d\times \mathcal{P}_2(\mathbb{R}^d)\to\mathbb{R}^d\) are Borel measurable. Here we use \(\mathcal{P}_2(\mathbb{R}^d)\) to denote the set of all square integrable probability measures on \(\mathbb{R}^d\).

By applying the weak convergence approach, we establish the large and moderate deviation principles. In addition, we obtain the central limit theorem and find the Volterra integral equation satisfied by the limiting process, which involves the Lions derivative of the drift coefficient.