Systems of variational inequalities for non-local operators related to optimal switching problems: Existence and uniqueness

TL;DR

This paper proves the existence and uniqueness of viscosity solutions for a system of non-local variational inequalities related to optimal switching problems driven by Levy processes, without sign restrictions on switching costs.

Contribution

It establishes a general comparison principle and proves existence and uniqueness of solutions for non-local operators in the context of Levy processes, with flexible switching costs.

Findings

Established a comparison principle for viscosity solutions.

Proved existence and uniqueness of solutions using Perron's method.

Allowed switching costs to depend on space and time without sign restrictions.

Abstract

In this paper we study a system of variational inequalities where the operator is non-local, possibly degenerate and of second order. A special case of this type of problem occurs in the context of optimal switching problems when the dynamics of the underlying state variables is described by an N-dimensional Levy process. We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the data. Using the comparison principle we then prove the existence of a unique viscosity solution to the system by Perron's method. Our main contribution is that we establish existence and uniqueness of viscosity solutions, in the setting of Levy processes and non-local operators, with no sign assumption on the switching costs and allowing them to depend on x as well as t.