On perpetual American put valuation and first-passage in a regime-switching model with jumps

In this paper we consider the problem of pricing a perpetual American put option in an exponential regime-switching L'{e}vy model. For the case of the (dense) class of phase-type jumps and finitely many regimes we derive an explicit expression for the value function. The solution of the corresponding first passage problem under a state-dependent level rests on a path transformation and a new matrix Wiener-Hopf factorization result for this class of processes.