A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection

TL;DR

This paper introduces a level-set method for controlling McKean-Vlasov dynamics with probabilistic state constraints, extending previous work to mean-field settings and applying it to renewable energy storage and portfolio optimization.

Contribution

It extends level-set control methods to mean-field problems with probabilistic constraints, including common noise, and demonstrates numerical solutions for energy storage and portfolio management.

Findings

Effective numerical method for constrained mean-field control problems.

Application to renewable energy storage with mean-field price impact.

Portfolio optimization with probabilistic wealth constraints.

Abstract

We consider the control of McKean-Vlasov dynamics (or mean-field control) with probabilistic state constraints. We rely on a level-set approach which provides a representation of the constrained problem in terms of an unconstrained one with exact penalization and running maximum or integral cost. The method is then extended to the common noise setting. Our work extends (Bokanowski, Picarelli, and Zidani, SIAM J. Control Optim. 54.5 (2016), pp. 2568--2593) and (Bokanowski, Picarelli, and Zidani, Appl. Math. Optim. 71 (2015), pp. 125--163) to a mean-field setting. The reformulation as an unconstrained problem is particularly suitable for the numerical resolution of the problem, that is achieved from an extension of a machine learning algorithm from (Carmona, Lauri{\`e}re, arXiv:1908.01613 to appear in Ann. Appl. Prob., 2019). A first application concerns the storage of renewable electricity in the presence of mean-field price impact and another one focuses on a mean-variance portfolio selection problem with probabilistic constraints on the wealth. We also illustrate our approach for a direct numerical resolution of the primal Markowitz continuous-time problem without relying on duality.