Optimal Switching in Finite Horizon under State Constraints

TL;DR

This paper investigates an optimal switching problem with state constraints, establishing the value function as a limit of penalized unconstrained problems and characterizing it via a variational inequality system.

Contribution

It introduces a novel approach to handle state constraints in switching problems by linking penalized problems to the constrained case and characterizing the value function as a maximal solution.

Findings

Value function is the limit of penalized unconstrained problems.

The value function solves a variational inequality system in the constrained viscosity sense.

Uniqueness of the solution does not hold; it is characterized as the maximal solution.

Abstract

We study an optimal switching problem with a state constraint: the controller is only allowed to choose strategies that keep the controlled diffusion in a closed domain. We prove that the value function associated with this problem is the limit of value functions associated with unconstrained switching problems with penalized coefficients, as the penalization parameter goes to infinity. This convergence allows to set a dynamic programming principle for the constrained switching problem. We then prove that the value function is a solution to a system of variational inequalities (SVI for short) in the constrained viscosity sense. We finally prove that uniqueness for our SVI cannot hold and we give a weaker characterization of the value function as the maximal solution to this SVI. All our results are obtained without any regularity assumption on the constraint domain.