Existence, uniqueness and regularity of solutions to systems of nonlocal obstacle problems related to optimal switching

TL;DR

This paper establishes existence, uniqueness, and regularity results for viscosity solutions to a system of nonlinear degenerate parabolic integro-differential equations with interconnected obstacles, relevant to optimal switching with Levy process dynamics.

Contribution

It generalizes previous results to more complex systems, providing new comparison principles, regularity estimates, and existence proofs for viscosity solutions.

Findings

Proved continuous dependence estimates for solutions.

Established comparison principle and uniqueness.

Constructed barrier functions for regularity and existence.

Abstract

We study viscosity solutions to a system of nonlinear degenerate parabolic partial integro-differential equations with interconnected obstacles. This type of problem occurs in the context of optimal switching problems when the dynamics of the underlying state variable is described by an $n$-dimensional Levy process. We first establish a continuous dependence estimate for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the partial integro-differential operator and on the obstacles and terminal conditions. Using the continuous dependence estimate, we obtain the comparison principle and uniqueness of viscosity solutions as well as Lipschitz regularity in the spatial variables. Our main contribution is construction of suitable families of viscosity sub- and supersolutions which we use as barrier functions to prove H\"older continuity in the time variable, and, through Perron's method, existence of a unique viscosity solution. This paper generalizes parts of the results of Biswas, Jakobsen and Karlsen (2010) and of Lundstr\"om, Nystr\"om and Olofsson (2014) to hold for more general systems of equations.