03-FEB-2025
We study the optimal stopping problem in continuous time, where an agent chooses a stopping time to maximise the expected value of a payoff process, following the classical framework of Snell (1952) and its continuous-time extension via the Doob–Meyer decomposition. The value function is characterised as the Snell envelope — the smallest supermartingale dominating the payoff — whose generator satisfies a Hamilton– Jacobi–Bellman variational inequality of obstacle type. The optimal stopping time is the first entry into the stopping region, where the value function equals the payoff, and the continuation region is determined by the strict inequality V > g. As a canonical application, we solve the perpetual American put option, obtaining the closed-form exercise boundary and value function, and illustrate how the exercise boundary moves with time-to-expiry under the finite-horizon formulation.