Mean field optimal stopping with uncontrolled state

We study a specific class of finite-horizon mean field optimal stopping problems by means of the dynamic programming approach. In particular, we consider problems where the state process is not affected by the stopping time. Such problems arise, for instance, in the pricing of American options when the underlying asset follows a McKean-Vlasov dynamics. Due to the time inconsistency of these problems, we provide a suitable reformulation of the original problem for which a dynamic programming principle can be established. To accomplish this, we first enlarge the state space and then introduce the so-called extended value function. We prove that the Snell envelope of the original problem can be written in terms of the extended value function, from which we can derive a characterization of the smallest optimal stopping time. On the enlarged space, we restore time-consistency and in particular establish a dynamic programming principle for the extended value function. Finally, by employing the notion of Lions measure derivative, we derive the associated Hamilton-Jacobi-Bellman equation, which turns out to be a second-order variational inequality on the product space $[0, T ] \times \mathbb{R}^d \times \mathcal{P}_2(\mathbb{R}^d)$; under suitable assumptions, we prove that the extended value function is a viscosity solution to this equation.