On the optimal stopping problem for diffusions and an approximation result for stopping times

In this article, we propose an alternative approach to studying the classical finite-horizon optimal stopping problem for multidimensional diffusions, which differs from the methods typically encountered in the literature. More specifically, the core of our method lies in a key equality for the value function, from which a series of well-known results easily follow. Indeed, this equality enables us to prove directly that the classical stopping time, at which the value function equals the terminal gain, is the smallest optimal stopping time, without resorting to the martingale approach and relying on the Snell envelope. Moreover, this equality allows us to rigorously demonstrate the dynamic programming principle, thus showing that the value function is the viscosity solution to the corresponding variational inequality. To prove this equality, we employ an approximation result for stopping times, which is of independent interest and can find application in other stochastic control problems involving stopping times, such as switching or impulsive problems, also of mean field type.