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

In this article, we study the classical finite-horizon optimal stopping problem for multidimensional diffusions through an approach that differs from what is typically found in the literature. More specifically, we first prove a key equality for the value function from which a series of results easily follow. This equality enables us to prove 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. Such an equality also shows that the value function does not change when the class of stopping times varies. To prove this equality, we use an approximation result for stopping times, which is of independent interest and can find application in other stochastic control problems involving stopping times, as switching or impulsive problems, also of mean field type.