A PDE approach to regularity of solutions to finite horizon optimal switching problems

We study optimal 2-switching and nn-switching problems and the corresponding system of variational inequalities. We obtain results on the existence of viscosity solutions for the 2-switching problem for various setups when the cost of switching is non-deterministic. For the nn-switching problem we obtain regularity results for the solutions of the variational inequalities. The solutions are C1,1C1,1-regular away for the free boundaries of the action sets.