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On the form of integral payoff in differential games with random duration

In the paper we consider a class of the differential games with random duration, hence with either the random terminal (T) or the initial (T0) time of the game. Within this context we investigate the problem of whether the integral payoff transformation used is most applications can be performed. We prove that the availability of such transformation is subject to a special condition on the utility function and the cumulative distribution function F(t). The presented result is extended in several directions. Two examples illustrating our theoretical results are presented.