Pricing of perpetual American options in diffusion models with progressively enlarged filtrations

We study the problems of pricing of perpetual American options in one-dimensional positive diffusion models of financial markets with progressively enlarged Brownian reference filtrations. It is assumed that the reward processes represent integrals of discounted continuous piecewise-linear functions of the current values of the asset price processes taken up to the stopping time of exercise with respect to the enlarged reference filtrations. The progressive enlargements of the initial Brownian filtrations are made by the first hitting times of the intensity processes with the rates depending on the current values of the underlying processes by independent exponentially distributed random variables. The optimal exercise times are shown to be the first hitting times by the intensity processes with the rates depending on the running values of the diffusion processes of certain independent exponentially distributed thresholds. The associated optimal stopping problems are reduced to equivalent coupled ordinary free-boundary problems which explicitly characterise the candidate thresholds as unique solutions to the associated arithmetic equations.