Existence, uniqueness and exponential ergodicity under Lyapunov conditions for McKean-Vlasov SDEs with Markovian switching

Weak convergence of Mckean-Vlasov stochastic differential equations with two-time-scale Markov switching

“…as n → ∞. Substituting ( 21), ( 22) and ( 23) into (19) yields (17) and (18) imply that I k 31 → 0, as n → ∞. Combining this estimation with ( 16) and (15) gives…”

“…It allows to describe the random environment and can handle numerous real-world applications in which discrete and continuous dynamics coexist. Nowadays, more and more researchers pay attention to study the McKean-Vlasov SDEs with Markov switching because of the widely application in economics, population, optimization and and other branches of sciences and engineering, see [3,17,21,22,26,27] and so on. For example, Wang [27] investigated the social optima of mean field linear-quadratic-Gaussian control models with Markov jump parameters, and the optimal social average cost is compared with the optimal individual cost in mean field games by virtue of the explicit expressions.…”

“…In [22], Nguyen considered a general stochastic maximum principle for Mean-Field controls with regime switching, and an example on conditional mean-variance portfolio selection with regime switching is given to illustrate the sufficient conditions for optimality. In [17], Liu established the uniqueness and exponential ergodicity under Lyapunov conditions for McKean-Vlasov SDEs with state-independent switching in a finite state space.…”

“…More applications about two-time-scale Markov chains can be seen in [32] and references therein. Motivated by [14,17], this paper considers the following McKean-Vlasov SDE with two-time-scale Markov switching…”

“…Proof. The existence and uniqueness of equation ( 1) can be found in [17,Theorem 3.1]. From equation (1), for p ≥ 2,…”