Systems of Integro-PDEs with Interconnected Obstacles and Multi-Modes Switching Problem Driven by L\'evy Process

TL;DR

This paper proves existence and uniqueness of solutions for a complex system of integro-PDEs with interconnected obstacles, linked to stochastic switching problems driven by Le9vy processes, using reflected BSDEs.

Contribution

It establishes the well-posedness of a new class of nonlinear integro-PDE systems related to stochastic switching with Le9vy noise, extending existing theory.

Findings

Existence and uniqueness of viscosity solutions for the system.

Continuity and uniqueness of the value function in the switching problem.

Application of reflected BSDEs with oblique reflection driven by Le9vy processes.

Abstract

In this paper we show existence and uniqueness of the solution in viscosity sense for a system of nonlinear $m$ variational integral-partial differential equations with interconnected obstacles whose coefficients $(f_i)_{i=1,\cdots, m}$ depend on $(u_j)_{j=1,\cdots,m}$. From the probabilistic point of view, this system is related to optimal stochastic switching problem when the noise is driven by a L\'evy process. The switching costs depend on $(t,x)$. As a by-product of the main result we obtain that the value function of the switching problem is continuous and unique solution of its associated Hamilton-Jacobi-Bellman system of equations. The main tool we used is the notion of systems of reflected BSDEs with oblique reflection driven by a L\'evy process.