A Differential Game with Random Time Horizon and Discontinuous Distribution

Abstract

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.

Time-Consistency of an Imputation in a Cooperative Hybrid Differential Game

“…This Section will build upon the results presented in [17]. We will skip most results that were previously reported except for those that are necessary for the understanding of the current material.…”

“…The optimization problem can be tackled using state parametrization under four continuous intervals (see [17] for details):…”

“…In this paper, we consider the problem of the sustainability of a cooperative solution in an n-person hybrid differential game. Earlier, a differential game with a random time horizon and discontinuous distribution was studied in [16], where only one jump was considered, and in [17], where the method of parametrization for calculation of the optimal controls was used. In this work we extend and generalize the results of [17] and put them in the context of hybrid optimal control.…”

The Hamilton–Jacobi–Bellman Equation for Differential Games with Composite Distribution of Random Time Horizon

“…In such problems, a composite distribution function for terminal time is used. By Zaremba et al in [6], it is assumed that the duration of the game is a random variable corresponding to the discontinuous cumulative distribution function. Gromov and Gromova [7,8] considered games with random horizon and composite distribution function for terminal time as hybrid differential games, since payoffs of players in such games take the form of the sums of integrals with different, but adjoint time intervals.…”