Continuous Dynamic Optimization Models with Discrete Events

One can gain important insights about how to apply continuous control measures in conventional optimal control problems and differential games. In such a framework it is usually assumed that the underlying system evolves continuously over time. However, such an approach might not always be realistic, as in many cases discrete events should be taken into account when determining the optimal strategy. Then the continuous control variable should not only be adapted before and after the event to anticipate and react to its consequences, but the control can also be used to influence the timing and/or the impact of the event. In particular, we consider the problem of a firm, which either can endogenously trigger such a discrete event by changing its business model at a certain point in time, or it is affected by it when its competitor changes its business model. The main goal of the present project is to apply and extend methods from multi-stage modeling and impulse control to study the implications of discrete events in an otherwise continuous framework. Preliminary calculations have shown that it can be optimal to switch back and forth between two alternating stages, a phenomenon which has found little attention in the multi-stage literature so far. We will: investigate, under which conditions such a phenomenon can occur; conduct sensitivity and bifurcation analyses; and we will study economic implications of such phenomena. Furthermore, we will examine the occurrence of periodic solutions arising due to the inclusion of impulse controls. Anticipation effects are of particular importance in multi-stage differential games, as players might be able to influence their competitor`s actions and as such the occurrence, timing, and impact of discrete events, like a change of business models. It is the aim of the present project to study the impact of the inclusion of discrete events and to consider the effects of symmetries and asymmetries between players.