On Poisson constrained control of linear diffusions

Abstract

The classical setting in optimal stopping and optimal control theory assumes that the agent controlling the system can operate continuously in time. In optimal stopping this setting is highly stylized for many applications, for example, in mathematical finance due to illiquid markets. In optimal stochastic control this setting often leads to optimal strategies being singular with respect to the Lebesgue measure, and thus the strategies are not feasible in practice. Hence, it is of importance to study these problems from such a perspective that their solutions are practically more implementable.

In this thesis we alter the classical setting by introducing an exogenous constraint, in the form of a signal process, for the control opportunities of the agent. In order to keep the problems more tractable, especially time-homogeneous and Markovian, the signal process is assumed to be a Poisson process with constant intensity. Consequently, the agent can only have influence on the system at discrete times. We call these control problems Poisson constrained control problems and study them when the dynamics are governed by linear diffusion processes.

Linear diffusions are particular enough to have a rich theory but still general enough to offer a class of interesting dynamics that are applicable in various situations. A key factor is also that many control problems with diffusions will lead to closed-form solutions. This thesis investigates to which extent the classical theory of diffusion can be applied in this class of control problems to form closed-form solutions.