Stochastic Optimal Multi-Modes Switching with a Viscosity Solution Approach

TL;DR

This paper studies the problem of optimal switching between multiple modes in a finite horizon, establishing the existence of optimal strategies and characterizing the value functions as viscosity solutions to a system of interconnected PDEs.

Contribution

It provides a new existence result for optimal strategies and proves the value functions are the unique viscosity solutions in the Markov process setting.

Findings

Existence of optimal switching strategies established.

Finite optimal strategies characterized via a verification theorem.

Value functions are the unique viscosity solutions to the PDE system.

Abstract

We consider the problem of optimal multi-modes switching in finite horizon, when the state of the system, including the switching cost functions are arbitrary ($g_{ij}(t,x)\geq 0$). We show existence of the optimal strategy, and give when the optimal strategy is finite via a verification theorem. Finally, when the state of the system is a markov process, we show that the vector of value functions of the optimal problem is the unique viscosity solution to the system of $m$ variational partial differential inequalities with inter-connected obstacles.