A viscosity solution as a piecewise classical solution to a free boundary problem for the optimal switching problem with simultaneous multiple switches

TL;DR

This paper demonstrates that under certain conditions, the viscosity solution to a free boundary problem in an optimal switching context can be explicitly constructed as a piecewise classical solution, enabling analytical solutions and region identification.

Contribution

It proves the viscosity solution can be represented as a series of piecewise classical solutions and provides an algorithm for computing free boundaries in optimal switching problems.

Findings

Explicit solution and free boundary identification for the problem in  Suzuki (2020)

Verification that the series of classical solutions equals the viscosity solution

Development of an algorithm to compute all free boundaries

Abstract

\citeN{suzuki2020optimal} proves the uniqueness of the viscosity solution to a variational inequality which is solved by the value function of the infinite horizon optimal switching problem with simultaneous multiple switchings. Although it also identifies each connected region possibly including at most one connected switching region, the exact switching regions of the solution are not identified. The problem is finally converted into a system of free boundary problems and generally solved by the numerical calculation. However, if the PDE part of the variational inequality has a classical solution, the viscosity solution may be constructed as a series of piecewise classical solutions, possibly analytical. Under a certain assumption we prove that the series of piecewise classical solutions is indeed the viscosity solution on $\real{}$, after we prove the smooth pasting condition is its necessary condition, and establish the algorithm to compute all the free boundaries. Applying the results to the concrete problem studied in \citeN{suzuki2020optimal} we find the explicit solution and identify the continuation and switching regions in a computer with Python programs.