Non-constant discounting and differential games with random time horizon

Previous results on non-constant discounting in continuous time are extended to the field of deterministic differential games with a stochastic terminal time. A dynamic programming equation is derived for problems with general time inconsistent preferences and random duration. Different cooperative and non-cooperative solution concepts for differential games with random duration are analyzed. The results are illustrated by solving the cake-eating problem describing the classical model of management of a nonrenewable resource.