## Discrete and Continuous Operational Calculus in Stochastic Games

##### Abstract

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 [1] 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).