Two explicitly solvable problems with discretionary stopping

This thesis is concerned with two explicitly solvable stochastic control problems that incorporate discretionary stopping. The first of these problems combines the features of the so-called monotone follower of singular stochastic control theory with optimal stop- ping. The uncontrolled state dynamics are modelled by a general one-dimensional It^o diffusion. The aim of the problem is to maximise the utility derived from the system's controlled state at the discretionary time when the system's control is terminated. This objective is re ected by an appropriate performance criterion, which also penalises con- trol expenditure as well as waiting. In the presence of rather general assumptions, the optimal strategy, which can take one of three qualitatively different forms, depending on the problem data, is fully characterised. The second problem is concerned with the optimal stopping of a diffusion with gen-eralised drift over an infinite horizon. The dynamics of the underlying state process are similar to the ones of a geometric Brownian motion. In particular, the drift of the state process incorporates the process' local time at a given level in an additive way. The ob- jective of this problem is to maximise the expected discounted payoff that stopping the underlying diffusion yields over all stopping times. The associated reward function is the one of a financial call option. The optimal stopping strategy can take six qualitatively different forms, depending on parameter values.