I am trying to formulate a stochastic sequential game in discrete time, and was trying to do so as is common in the literature but couldn't find the appropriate setting anywhere. I'm sure that it exists somewhere but the body of work on stochastic games is so huge that every time I'm looking at something the setting is different and so far I haven't found the correct setting; I am thinking of giving up and invent everything by myself, which is probably a bad idea. So, if anyone knows a book or a paper where this is formulated, I will be grateful.
My setting is as follows. There is a dynamic system whose state is controlled by noise, the actions of a maximizer and the actions of a minimizer. The actions of the maximizer are a stochastic sequence adapted to some filtration $\{{\cal F}_t\}$. After each time where the maximizer acts, the noise changes the state of the system and then the minimizer acts. Then the maximizer acts again and so the game progresses.
I am not sure about how to define the strategy of the minimizer. I thought about defining it as a deterministic mapping of the maximizer's control and the system's state that is non-anticipating (or causal). Another idea would be to define it as a stochastic process adapted to some filtration. Perhaps these two suggestions coincide but I am not sure about this.
As I said before, I guess that I am not the first one to think of a problem with this setting but I had no luck so far with finding a reference. The closest setting I could find was in Bardi-Capuzzo-Dolcetta's book on Optimal Control and Viscosity Solutions in the chapter on differential games where they discuss discrete time approximations to differential games, but there is no stochasticity there.
Any help is appreciated, thanks a lot!
You could look at the tug-of-war games for the p-Laplacian, introduced in the paper below. There are many followup papers to this.
https://arxiv.org/abs/math/0607761
It's rather specific, but is a min/max stochastic discrete time game. I'm not sure of a general reference for this kind of thing.