The Optimal Stopping Problem under a Random Horizon

This paper considers a pair (F,τ), where F is a filtration representing the “public” flow of information that is available to all agents over time, and τ is a random time that might not be an F-stopping time. This setting covers the case of a credit risk framework, where τ models the default time of a firm or client, and the setting of life insurance, where τ is the death time of an agent. It is clear that random times cannot be observed before their occurrence. Thus, the larger filtration, G, which incorporates F and makes τ observable, results from the progressive enlargement of F with τ. For this informational setting, governed by G, we analyze the optimal stopping problem in three main directions. The first direction consists of characterizing the existence of the solution to this problem in terms of F-observable processes. The second direction lies in deriving the mathematical structures of the value process of this control problem, while the third direction singles out the associated optimal stopping problem under F. These three aspects allow us to deeply quantify how τ impacts the optimal stopping problem and are also vital for studying reflected backward stochastic differential equations that arise naturally from pricing and hedging of vulnerable claims.