Mean-field Models Involving Continuous-state-dependent Random Switching: Nonnegativity Constraints, Moment Bounds, and Two-time-scale Limits

Abstract

This work concerns mean-field models, which are formulated using stochastic differential equations. Different from the existing formulations, a random switching process is added. The switching process can be used to describe the random environment and other stochastic factors that cannot be explained in the usual diffusion models. The added switching component makes the formulation more realistic, but it adds difficulty in analyzing the underlying processes. Several properties of the mean-field models are provided including regularity, nonnegativity, finite moments, and continuity. In addition, the paper addresses the issue when the switching takes place an order of magnitude faster than that of the continuous state. It derives a limit that is an average with respect to the invariant measure of the switching process.