2025 Theses Doctoral

Some Quandaries in the Theory of Optimal Stopping

Groenewald, Richard Andrew Umberto

The primary objective of this dissertation is to study and derive the properties of several optimal stopping problems, and it is split into three parts.

In the first part, we use the dual perspective for optimal stopping, introduced by Davis and Karatzas, to derive pathwise conditions for commonly sought properties of a large class of optimal stopping problems, which depend on a parameter taking values in an arbitrary topological space. In particular, we show that the continuity, with respect to this parameter, of the value function, of the optimal stopping time, and of the spatial derivative of the value function, can be derived from the path properties of the reward process and its Snell envelope. Furthermore, we show that these properties imply the existence of Nash equilibria in a large class of stopping games. We provide several illustrative examples and applications. Of related, and potentially independent interest, is the continuity of solutions to stochastic differential equations with respect to a parameter other than time, and we also address this topic for one dimensional diffusions.

In the second part, we study a sequential estimation problem for an unknown reward in the presence of a random horizon. The reward takes one of two predetermined values which can be inferred from the drift of a Wiener process, which serves as a signal. The objective is to use the information in the signal to estimate the reward which is made available until a stochastic deadline that \emph{depends} on its value. The observer must therefore work quickly to determine if the reward is favorable and claim it before the deadline passes. Under general assumptions on the stochastic deadline (which may in be replaced by general discount functions), we provide a full characterization of the solution that includes an identification with the unique solution to a free-boundary problem. Our analysis derives regularity properties of the solution that imply its ``smooth fit'' with the boundary data, and we show that the free-boundary solves a particular integral equation. The continuity of the free-boundary is also established under additional structural assumptions that lead to its representation in terms of a continuous transformation of a monotone function. We provide illustrations for several examples of interest.

In the third part, we analyze a two-player, nonzero-sum Dynkin game of stopping with incomplete information. We assume that each player observes his own Brownian motion, which is not only independent of the other player's Brownian motion but also not observable by the other player. The player who stops first receives a payoff that depends on the stopping position. Under appropriate growth conditions on the reward function, we show that there are infinitely many Nash equilibria in which both players attain infinite expected payoffs. In contrast, the only equilibrium with finite expected payoffs mandates immediate stopping by at least one of the players. Our results hold in the settings of both pure and mixed strategies.