Conditional McKean-Vlasov SDEs with jumps and Markovian regime-switching: wellposedness, propagation of chaos, averaging principle

Abstract

We investigate the conditional McKean-Vlasov stochastic differential equations with jumps and Markovian regime-switching. We establish the strong wellposedness using L 2 -Wasser-stein distance on the Wasserstein space. Also, we establish the propagation of chaos for the associated mean-field interaction particle system with common noise and provide an explicit bound on the convergence rate. Furthermore, an averaging principle is established for two time-scale conditional McKean-Vlasov equations, where much attention is paid to the convergence of the conditional distribution term.

Comparison of Difficulty Levels of Examples in High School Mathematics Textbooks

“…In contrast to McKean-Vlasov SDEs without common noise, the research on McKean-Vlasov SDEs with common noise is not too rich. Yet, in the past few years, there are still some progresses on qualitative and quantitative analyses; see, for example, [3,22,37] on well-posedness, and [6,16,23,37] concerned with finite-time conditional propagation of chaos. According to [6, p. 110-112], the random distribution flow (µ t ) t>0 associated with (1.3) solves the nonlinear FPE:…”

Coupling by Change of Measure for Conditional McKean-Vlasov SDEs and Applications

“…When there exists a common noise in the mean field particle system, the limit equation of a single particle becomes a conditional distribution dependent stochastic differential equation, which is called conditional McKean-Vlasov stochastic differential equation and the conditional distribution of the solution with respect to the common noise as a probability measure-valued process solves stochastic nonlinear Fokker-Planck-Kolmogorov equation. Compared with the McKean-Vlasov SDEs, there are fewer results on conditional McKean-Vlasov ones, one can refer to [1,3,4,5,10,2,13,7,16,17,21] for well-posedness, [13,7,12] for the study of stochastic nonlinear Fokker-Planck-Kolmogorov equation and [2,6,8,12,18,1,16,17] for conditional propagation of chaos.…”