OPTIMAL CONTROL WITH A STOCHASTIC SWITCHING TIME: INTRODUCTION AND SOLUTION APPROACHES
When planning an optimal policy, a farsighted decision-maker should account for the possibile occurrence of disruptive events over the course of the time horizon. For example, when planning the optimal emission abatement policy, account for a possible climate catastrophe; when planning industrial production, account for an unpredictable disruption that may affect the producer’s profit.
In the optimal control framework, a stochastic switching time is a random instant, modeled as a positive random variable, which marks a regime shift – i.e., an abrupt and irreversible change in the system – which splits the planning horizon into two stages. The shift may affect the payoff and/or the state trajectory in several ways, all of which are included in the analysis of the most general scenario. In search for the optimal policy under this kind of uncertainty, two methods are featured in the literature: the “backward” approach and the “heterogeneous” one. The two approaches will be described, compared, and then applied to a marketing toy model.