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

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.