A Differential Game with Random Time Horizon and Discontinuous Distribution

One class of differential games with random duration is considered. It is assumed that the duration of the game is a random variable with values from a given finite interval. The cumulative distribution function (CDF) of this random variable is assumed to be discontinuous with two jumps on the interval. It follows that the player's payoff takes the form of the sum of integrals with different but adjoint time intervals. In addition, the first interval corresponds to the zero probability of the game to be finished, which results in terminal payoff on this interval. The method of construction optimal solution for the cooperative scenario of such games is proposed. The results are illustrated by the example of differential game of investment in the public stock of knowledge.