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.
Highlights
Dynamic processes with many participants are well described by the differential game theory framework
This type of models was initially introduced in [6], where a differential game of pursuit with terminal payoff and stochastic terminal time was considered; the first problem formulation with integral payoff function and random time horizon with continuous distribution function was considered in [7] and later extended to the case of differential game with random time horizon and discounting in [8]
Assume that the game starts at the initial moment t0 and the initial state x0 ; the duration of the game is a random variable such that it corresponds to the particular cumulative distribution function (CDF) described in the below notation
Summary
Dynamic processes with many participants are well described by the differential game theory framework (e.g., for differential games see [1], [2] in Russian, and their applications in economics in [3]). One class of differential games with a random time horizon is considered. Particular classes of discontinuous models had been formulated in [9] for the case with only one jump and in [10] for the step form of the CDF. For a class of differential games with random time horizon, with continuous but composite CDF, further generalized in [11]. A method to construct optimal controls for the introduced class of games with discontiniuous CDF is proposed This method refers to the idea of defining the connecting trajectory points at the edges of intervals as numeric parameters and usage of the maximum Pontryagin principle [16], after every interval. The last Section presents a numeric example for the stock investment model
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
