November 14, 2024Open Access

Stochastic optimal transport and Hamilton–Jacobi–Bellman equations on the set of probability measures

Key Points

Existence and uniqueness of viscosity solutions are established, ensuring reliability in interpretations of stochastic optimal transport.
The analysis reveals general comparison principles, particularly addressing stochastic elements in optimal control problems.
Framework utilizes viscosity solutions of the Hamilton–Jacobi–Bellman equation on probabilistic measures to model optimal transport dynamics effectively in stochastic settings, enhancing viability of solutions for complex systems of measures throughout various scenarios. Through a novel framework for viscosity solutions, we clarify the interplay between controllability and optimal transport, allowing for advanced applications in stochastic environments.

Abstract

We introduce a stochastic version of the optimal transport problem. We provide an analysis by means of the study of the associated Hamilton–Jacobi–Bellman equation, which is set on the set of probability measures. We introduce a new definition of viscosity solutions of this equation, which yields general comparison principles, in particular for cases involving terms modeling stochasticity in the optimal control problem. We are then able to establish results of existence and uniqueness of viscosity solutions of the Hamilton–Jacobi–Bellman equation. These results rely on controllability results for stochastic optimal transport that we also establish.

Stochastic optimal transport and Hamilton–Jacobi–Bellman equations on the set of probability measures

Key Points

Abstract

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