The Hamilton-Jacobi-Bellman equation for a class of differential games with random duration

We consider the class of differential games with random duration. We show that a problem with random game duration can be reduced to a standard problem with an infinite time horizon. A Hamilton-Jacobi-Bellman-type equation is derived for finding optimal solutions in differential games with random duration. Results are illustrated by an example of a game-theoretic model of nonrenewable resource extraction. The problem is analyzed under the assumption of Weibull-distributed random terminal time of the game.