Viscosity solutions for controlled McKean--Vlasov jump-diffusions

TL;DR

This paper develops a new intrinsic notion of viscosity solutions for a class of non-linear integro-differential equations on Wasserstein space, related to optimal control of McKean--Vlasov jump-diffusions, and proves their uniqueness.

Contribution

It introduces an intrinsic approach to viscosity solutions on Wasserstein space and establishes a comparison theorem and uniqueness for the value function.

Findings

Established a comparison theorem for viscosity solutions

Proved the value function is the unique viscosity solution

Developed an intrinsic notion not relying on Hilbert space lifting

Abstract

We study a class of non linear integro-differential equations on the Wasserstein space related to the optimal control of McKean--Vlasov jump-diffusions. We develop an intrinsic notion of viscosity solutions that does not rely on the lifting to an Hilbert space and prove a comparison theorem for these solutions. We also show that the value function is the unique viscosity solution.