McKean-Vlasov forward-backward doubly stochastic differential equations and applications to stochastic control

2024/06/12 by Al-Hussein, AbdulRahman, Ninouh, Abdelhakim, Gherbal, Boulakhras
#60H10 #93E20 #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR)

paper · doi:10.48550/arxiv.2406.07881

Abstract

This paper investigates first the existence and uniqueness of solutions for McKean-Vlasov forward-backward doubly stochastic differential equations (MV-FBDSDEs) in infinite-dimensional real separable Hilbert spaces. These equations combine the features of forward-backward doubly stochastic differential equations with the mean-field approach, allowing the coefficients to depend on the solution distribution. We establish the existence and uniqueness of solutions for MV-FBDSDEs using the method of continuation and provide an example and a counterexample to illustrate our findings. Moreover, we extend the practical applicability of our results by employing them within the context of the stochastic maximum principle for a control problem governed by MV-FBDSDEs. This study contributes to the field of stochastic control problems and presents the first analysis of MV-FBDSDEs in infinite-dimensional spaces.

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