On Problems of optimal timing under regime switching and poisson constraints

Abstract

The focus of this thesis is in analyzing optimal stopping (OSP) and impulse control problems (ICP) of linear diffusions under regime switching and Poisson constraints. Regime switching means that the problem’s parameters undergo a change at a random instant. Regime switching models are useful for modelling problems where time horizon is long compared to the expected duration of relevant exogenous conditions such as business cycles or macroeconomic policies. We study OSPs and ICPs in the presence of a regime switch in a single paper. The other three papers are devoted to various timing problems under a Poisson constraint.

The Poisson constraint can be explained as follows. In the standard formulations of various stochastic control problems, controls can be exercised at any time. This is not always a realistic assumption as in practice there may exist liquidity and in- formation constraints that pose significant limitations to the available controlling opportunities. We model these limitations by constraining the admissible control times to be the set of jump times of an independent time-homogeneous Poisson process.

The underlying processes of the problems are assumed to be linear diffusions. The reason for this is that linear diffusions admit a particularly rich analytical theory which allows us to obtain explicit information about the structure of solutions in addition to the usual abstract existence and uniqueness results. We illustrate the general theory with examples ranging from economics and mathematical finance to operations research and optimal harvesting of renewable resources.