A level-set approach for stochastic optimal control problems under controlled-loss constraints

TL;DR

This paper introduces a level-set method to solve stochastic optimal control problems with multiple controlled-loss constraints at different dates, avoiding strong assumptions on process dynamics.

Contribution

It develops a novel level-set approach combined with exact penalization to handle state constraints in stochastic control without restrictive assumptions.

Findings

Provides a new characterization of the value function via Hamilton-Jacobi-Bellman equations.

Enables handling of complex constraints through an innovative state-constrained reformulation.

Offers a practical method for stochastic control problems with multiple loss constraints.

Abstract

We study a family of optimal control problems under a set of controlled-loss constraints holding at different deterministic dates. The characterization of the associated value function by a Hamilton-Jacobi-Bellman equation usually calls for additional strong assumptions on the dynamics of the processes involved and the set of constraints. To treat this problem in absence of those assumptions, we first convert it into a state-constrained stochastic target problem and then apply a level-set approach. With this approach, the state constraints can be managed through an exact penalization technique.