Ergodic distribution dependent BSDE and application to long-time behavior of finite horizon distribution dependent BSDE
Kaplan Desbouis, Adrien Richou
TL;DR
The paper investigates ergodic distribution-dependent BSDEs associated with MV-SDEs under two dissipativity regimes, establishing existence, uniqueness, and regularity of the ergodic solution and its decoupled representations. It introduces a discounted-approximation approach to derive the ergodic constant and shows exponential long-time convergence for finite-horizon MV-BSDEs, linking to ergodic MV-HJB equations. These results are then applied to a partial McKean–Vlasov ergodic control problem, deriving the ergodic HJB framework and demonstrating the existence of optimal controls with convergence properties for the value function. The work broadens the understanding of long-time behavior in distribution-dependent settings and provides tools for ergodic control in mean-field-type dynamics.
Abstract
After proving existence and uniqueness of ergodic distribution dependent backward stochastic differential equations (BSDEs) under strong and weak dissipativity regimes for the underlying McKean--Vlasov SDE, we leverage this new framework to investigate the long-time behavior of distribution dependent BSDEs on a finite-time horizon. Finally, we apply our results to solve an ergodic McKean--Vlasov stochastic control problem and study the long-time behavior of the value function of a finite-horizon McKean--Vlasov stochastic control problem.