Viscosity Solutions for a System of PDEs and Optimal Switching

TL;DR

This paper investigates the optimal switching problem with arbitrary costs, establishing existence, uniqueness, and characterization of solutions via viscosity solutions for a system of interconnected PDEs, relevant for economic and environmental applications.

Contribution

It introduces a novel approach to solve the m-states optimal switching problem with arbitrary costs using viscosity solutions and approximation schemes.

Findings

Existence of optimal strategies proven via verification theorem.

Uniqueness and characterization of value processes as viscosity solutions.

Value functions are deterministic and solve a system of variational PDEs.

Abstract

In this paper, we study the $m$-states optimal switching problem in finite horizon, when the switching cost functions are arbitrary and can be positive or negative. This has an economic incentive in terms of central evaluation in cases where such organizations or state grants or financial assistance to power plants that promotes green energy in their production activity or what uses less polluting modes in their production. We show existence for optimal strategy via a verification theorem then we show existence and uniqueness of the value processes by using an approximation scheme. In the markovian framework we show that the value processes can be characterized in terms of deterministic continuous functions of the state of the process. Those latter functions are the unique viscosity solutions for a system of $m$ variational partial differential inequalities with inter-connected obstacles.