On the optimal stopping problem for diffusions and an approximation result for stopping times
Andrea Cosso, Laura Perelli
TL;DR
The paper tackles the finite-horizon optimal stopping problem for multidimensional diffusions and derives a key equality for the value function that allows bypassing the Snell envelope in identifying the smallest optimal stopping time; it then shows the dynamic programming principle and viscosity solution property of the value function to the associated variational inequality. Central to the analysis is the equality $\tilde{v}(\theta,\xi) = \esssup_{\tau\in \mathcal{T}_{\theta,T}} E[ \int_{\theta}^{\tau} f(s,X^{\theta,\xi}_s) ds + g(X^{\theta,\xi}_\tau) | \mathcal{F}_\theta]$, which yields $v=\tilde{v}$ and characterizes the optimal stopping time as $\hat{\tau}^{t,x}=\inf\{ s: v(s,X^{t,x}_s)=g(X^{t,x}_s)\}$. The work also proves a dynamic programming principle and provides a rigorous viscosity-solution framework for the corresponding Hamilton-Jacobi-Bellman equation, together with an independent approximation result for stopping times (Theorem \ref{T:Approx}) that facilitates applications to switching, impulsive, and mean-field-type problems. The combination of a non-martingale route to optimal stopping and the explicit constructive approximation of stopping times broadens the toolkit for stochastic control problems with stopping decisions and strengthens connections to variational inequalities and mean-field control.
Abstract
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.