2026 Theses Doctoral

Explicitly Solvable Problems in Optimal Stopping, Stochastic Control, and Sequential Inference

Gaitsgori, Georgy

This dissertation is devoted to solving explicitly three problems at the intersection of optimal stopping, stochastic control, and sequential inference.

The first chapter after the introduction (based on joint work with Václav E. Beneš and Ioannis Karatzas considers stochastic control with discretionary stopping for the drift of a diffusion process over an infinite time horizon. The objective is to choose a control process and a stopping time to minimize the expectation of a convex terminal cost in the presence of a fixed operating cost and a control-dependent running cost per unit of elapsed time. Under appropriate conditions on the coefficients of the controlled diffusion, an optimal pair of control and stopping rules is shown to exist. Moreover, under the same assumptions, it is shown that the optimal control is a constant which can be computed fairly explicitly; and that it is optimal to stop the first time an appropriate interval is visited.

We consider also a constrained version of the above problem, in which an upper bound on the expectation of available stopping times is imposed; we show that this constrained problem can be reduced to an unconstrained problem with some appropriate change of parameters and, as a result, solved by similar arguments.

The second research chapter (based on joint work with Steven Campbell and Richard Groenewald formulates a controlled version of the Bayesian sequential testing problem for the drift of a Wiener process, in which the observer exercises discretion over the signal intensity. This control incurs a running cost that reflects the resource demands of information acquisition. The objective is to minimize the total expected cost, combining both the expenditure on control and the loss from misclassifying the unknown drift. By allowing for a general class of loss functions and 𝘢𝘯𝘺 𝘮𝘦𝘢𝘴𝘶𝘳𝘢𝘣𝘭𝘦 cost of control, our analysis captures a broad range of sequential inference problems.

We show that when a function, determined by the cost structure, admits a global minimizer, the optimal control is constant and explicitly computable, thereby reducing our setting to a solvable optimal stopping problem. If no such minimizer exists, an optimal control does not exist either, yet the value function remains explicit. Our results thus demonstrate that full tractability can be retained even when extending sequential inference to include endogenous control over the information flow.

The third research chapter (based on joint work with Richard Groenewald) studies 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.