Two Different Solution Techniques for an Optimal Control Problem with a Stochastic Switching Time

In optimal control theory, strategic decision making requires the consideration of unforeseen disruptions that may arise within a predetermined time horizon. In this context, we introduce the concept of "stochastic switching time" as a random moment in time at which a sudden, irreversible alteration takes place in the system's dynamics or in the payoff function. To address optimal decision-making under such uncertain conditions, the literature presents two prominent methodologies: The "backward" approach and the "heterogeneous" approach. In this study, we offer an exposition and a comparative analysis of these two approaches. Finally, we present an illustrative example to show, in a detailed context, the advantages and disadvantages associated with these two solution strategies.