Coupled McKean-Vlasov stochastic differential equations with jumps

TL;DR

This paper studies coupled McKean-Vlasov stochastic differential equations with jumps, establishing superposition principles, well-posedness conditions, and ergodic properties, including exponential ergodicity, for these complex stochastic systems.

Contribution

It introduces superposition principles and well-posedness conditions for coupled MVSDEs with jumps, and investigates their ergodic behavior, including exponential ergodicity.

Findings

Superposition principles for coupled MVSDEs with jumps established

Conditions for well-posedness of solutions provided

Exponential ergodicity demonstrated for certain MVSDEs with jumps

Abstract

This work concerns a type of coupled McKean-Vlasov stochastic differential equations (MVSDEs in short) with jumps. First, we prove superposition principles for these coupled MVSDEs with jumps and non-local space-distribution dependent Fokker-Planck equations. Since superposition principles are related to the well-posedness of weak solutions for coupled MVSDEs with jumps, then we give some conditions to assure it. After this, we construct space-distribution valued Markov processes associated with these coupled MVSDEs with jumps. Finally, the ergodicity of these coupled MVSDEs with jumps are investigated. As a by-product, we show the exponential ergodicity for a type of MVSDEs with jumps.