On the existence of solutions of unbounded optimal stopping problems
The aim of the paper is to give sufficient conditions for the existence of solutions of optimal stopping problems for strong Markov processes when the payoff function is unbounded. This is a novelty: the previous literature has dealt only with problems where the payoff function was bounded in some sense. Moreover, the authors show that the solution can be represented as the first hitting time of the observable process. To be more concrete, consider the optimal stopping problem \[ V(x) \;= \;\sup_{\tau \in \mathcal{M}} \mathbb{E}_x G(X_\tau), \quad s \in S, \] where \(\mathcal{M}\) denotes the set of Markov times, \(S\) is a phase space and \(G:S \to \bar{\mathbb{R}}\) is the payoff function. The assumptions ``\(G\) is bounded from below'' and ``\(\mathbb{E}_x \sup_{t\geq 0} |G(X_t)| < \infty\) for any \(x \in S\)'' are replaced by a weaker condition and the optimal stopping time is given explicitly. Moreover, the authors generalize their main result to the case where the Markov process is not homogeneous or the payoff function \(G\) depends on the time by reducing the problem to one with a homogeneous process. Again, they give weak conditions under which there exists a solution. Lastly, the authors consider an optimal stopping problem for a Brownian motion and show by applying the previous results that there also exists a solution.
scientific article; zbMATH DE number 6473893