Andi Bodnariu: Stochastic optimal control and stopping, games and time inconsistency

Abstract: This thesis contributes to the field of stochastic optimisation considering a version of a dividend problem, a stochastic differential game with incomplete information, as well as game-theoretic approaches to time-inconsistent stopping and control. The content of this thesis is based upon four papers.

Paper I considers a game theoretic approach to a time-inconsistent stopping problem, where the time-inconsistency is due to non-exponential discounting. We introduce a novel class of mixed stopping strategies and provide a verification theorem. Furthermore, we consider an example, where there is no equilibrium when using only pure stopping times. In this case we are able to construct an equilibrium in the class of mixed stopping times.

Paper II considers a continuous time stochastic controller stopper game with incomplete information. The stopper can be seen as owner of an asset and a controller as the manager who is either effective or non-effective. The manager earns a salary paid by the owner. An effective manager can choose to exert effort at a cost in order to increase the drift of the asset while a non-effective manager cannot act. The owner can choose to stop the game at any time based on observations of the movement of the asset. The owner is not able to observe directly whether the manager is effective or non-effective, making this a game of incomplete information. We derive a Nash equilibrium to this game, given as a threshold solution depending on the conditional probability that the manager is effective.

Paper III considers a time-inconsistent singular stochastic control problem, where the time-inconsistency is due to non-exponential discounting. We introduce a new class of "mild" threshold controls, which are given by an exploding rate that generates an inaccessible boundary for the underlying diffusion. These "mild" controls stand in contrast to the "strong" threshold controls that have been considered previously and amount to a Skorokhod reflection at an upper boundary. We provide an appropriate equilibrium condition for these controls as well as a verification theorem. Furthermore we provide an example, where no equilibrium exist if we only consider "strong" threshold strategies. We are, however able to find an equilibrium when considering "mild" threshold control strategies.

Paper IV considers a dividend problem with ruin at zero surplus or if the surplus spends too long time below a certain threshold of distress. We completely solve the problem considering three different cases. If the distress threshold is small or large the optimal control results in paying out dividends above a certain threshold. If the distress threshold takes intermediate values, the optimal control results in paying out dividends in two separated regions.

Collectively, these results advance the theory for optimal stochastic control and stopping, by enriching the literature with new problems, as well as presenting solution structures that have not been considered previously.