McKean-Vlasov multivalued stochastic differential equations with oblique subgradients and related stochastic control problems

TL;DR

This paper establishes existence and uniqueness results for McKean-Vlasov multivalued stochastic differential equations with oblique subgradients, and applies these to stochastic control problems with convex constraints.

Contribution

It introduces new techniques to handle multivalued McKean-Vlasov SDEs with oblique subgradients without relying on maximal monotony, and connects these to stochastic control.

Findings

Proved existence of weak solutions.

Proved existence and uniqueness of strong solutions.

Established dynamic programming principle for the control problem.

Abstract

In this article, we prove the existence of weak solutions as well as the existence and uniqueness of strong solutions for McKean-Vlasov multivalued stochastic differential equations with oblique subgradients (MVMSDEswOS, for short) by means of the equations of Euler type and Skorohod's representation theorem. For this type of equation, compared with the method in [19,13], since we can't use the maximal monotony property of its constituent subdifferential operator, some different specific techniques are applied to solve our problems. Afterwards, we give an example for MVMSDEswOS with time-dependent convex constraints, which can be reduced to MVMSDEswOS. Finally, we consider an optimal control problem and establish the dynamic programming principle for the value function.