Discrete and Continuous Operational Calculus in Stochastic Games
Iwezulu, Kenneth Ibe
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First, we consider a class of antagonistic stochastic games between two players A and B. The game is specified in terms of two "hostile" stochastic processes representing mutual attacks upon random times exerting casualties of random magnitudes. This game is observed upon random epochs of time and the outcome of the game is not known in real time. The game ends at the time when the underlying fixed threshold of either player is crossed (referred to as the first passage time). The first passage time is then shifted to an epoch, i.e. upon one of the observation instants of time. Thus, the narrative of the game is delayed allowing the players to continue fighting each other beyond their assumed merits of endurance. We target the first passage time of the defeat and the amount of casualties to either player upon the end of the game. Here we validate our claim of analytic tractability of the general formulas obtained in  under various transforms. We also consider a class of antagonistic stochastic games in real time between two players A and B formalized by two marked point processes. The players attack each other at random times with random impacts. Either player can sustain casualties up to a fixed threshold. A player is defeated when its underlying threshold is crossed. Upon that time (referred to as the first passage time), the game is over. We introduce a joint functional of the first passage, along with the status of each player upon this time, meaning the cumulative magnitude of casualties to each player upon the end of the game, obtained in an analytically tractable form. We then use discrete and continuous operational calculus for the transform inversion. We demonstrate that in a special case that the discrete operational calculus is more efficient allowing us to avoid numerical inversion. It leads to totally explicit formulas for the joint distribution of associated random variables (first passage time and the status of cumulative casualties to the players upon the end of the game).