Systems of quasi-variational inequalities related to the switching problem

TL;DR

This paper establishes the existence of solutions for a system of quasi-variational inequalities linked to switching problems, using stochastic methods and reflected BSDEs, and demonstrates the existence of optimal strategies.

Contribution

It introduces a novel approach connecting quasi-variational inequalities with reflected BSDEs for switching problems involving measure data.

Findings

Existence of weak solutions for the system of inequalities.

Representation of solutions via reflected BSDEs.

Existence of optimal switching strategies.

Abstract

We prove the existence of weak solution for a system of quasi-variational inequalities related to a switching problem with dynamic driven by operator associated with a semi-Dirichlet form and with measure data. We give a stochastic representation of solutions in terms of solutions of a system of reflected BSDEs with oblique reflection. As a by-product, we prove the existence of an optimal strategy in the switching problem and show regularity of the payoff function.