Yihao Sheng | Search arXiv

arXiv2026

On Fast-Slow Mean-Field Forward-Backward Stochastic Systems

We establish an averaging principle for a class of multiscale mean-field forward-backward stochastic differential equations and identify several novel phenomena that are absent from classical fast-slow systems. In contrast with classical fast-slow systems, the effective dynamics cannot in general be obtained by simply freezing deterministic slow parameters and averaging against the invariant measure of the resulting fast equation. The appropriate averaging object is instead provided by a frozen fast dynamics in a random environment and its associated conditional invariant measures, which retain the coupling between the slow state and its distribution. The forward-backward structure creates a further obstruction: local averaging estimates need not remain stable when propagated over an arbitrary time horizon. We identify a uniform restart stability condition for the averaged system under which this obstruction can be overcome. Using a joint lifted semigroup for the state-law dynamics, together with a two-scale discretization and a Gordin-type decomposition, we prove strong averaging for both the forward and backward components with optimal convergence rate $O(\varepsilon^{1/2})$. As an application, we apply the general theory to a class of mean-field stochastic control problems and develop an efficient algorithm for solving such mean-field control problems.

math.OC↗

arXiv2023

Continuous Ergodic Capacities

The objective of this paper is to characterize the structure of the set $Θ$ for a continuous ergodic upper probability $\mathbb{V}=\sup_{P\inΘ}P$ (Theorem \ref {main result}): . $Θ$ contains a finite number of ergodic probabilities; . Any invariant probability in $Θ$ is a convex combination of those ergodic ones in $Θ$; . Any probability in $Θ$ coincides with an invariant one in $Θ$ on the invariant $σ$-algebra. The last property has already been obtained in \textsl{Cerreia-Vioglio, Maccheroni, and Marinacci} \cite{ergodictheorem}, which firstly studied the ergodicity of such capacities. As an application of the characterization, we prove an ergodicity result (Theorem \ref {improve}), which improves the result in \cite{ergodictheorem} in the sense that the limit of the time mean of $ξ$ is bounded by the upper expectation $\sup_{P\inΘ}E_P[ξ]$, instead of the Choquet integral. Generally, the former is strictly smaller.

math.PR↗