This paper deals with a class of stochastic differential games where the mode of play changes according to a stochastic jumpprocess. Between two successive random jump times the process is either fully deterministic or is represented as a diffusion process obeying an Ito equation. In the piecewise deterministic case one considers the class of piecewise open-loop strategies. Each player uses an open-loop control which can be modified and adapted to the current state observation only at jump times. One characterizes equilibria for this class of games and shows on a duopoly model with technological changes how this modeling framework can be used in economic analysis. For the piecewise diffusion case one considers feedback strategies. Equilibria are characterized via the Hamilton-Jacobi-Bellman equations. One then shows how the formalism of piecewise diffusion games can be used in the construction of cooperative equilibria à la Porter. An example dealing with exploitation of a renewable resource by two economic agents illustrates this theory.
Published January 1990 , 19 pages