Optimal Multi-Modes Switching Problem in Infinite Horizon

TL;DR

This paper addresses the deterministic multi-modes switching problem over an infinite horizon, employing probabilistic and viscosity solutions methods to establish the uniqueness of the value functions in a Markovian setting.

Contribution

It introduces a novel approach combining probabilistic tools and viscosity solutions to analyze the infinite-horizon multi-modes switching problem with arbitrary costs.

Findings

Proves the value functions are the unique viscosity solutions to the system of variational inequalities.

Establishes the connection between the switching problem and firm valuation in financial markets.

Develops a probabilistic framework for solving complex impulse control problems.

Abstract

This paper studies the problem of the deterministic version of the Verification Theorem for the optimal m-states switching in infinite horizon under Markovian framework with arbitrary switching cost functions. The problem is formulated as an extended impulse control problem and solved by means of probabilistic tools such as the Snell envelop of processes and reflected backward stochastic differential equations. A viscosity solutions approach is employed to carry out a finne analysis on the associated system of m variational inequalities with inter-connected obstacles. We show that the vector of value functions of the optimal problem is the unique viscosity solution to the system. This problem is in relation with the valuation of firms in a financial market.