Asymptotic Perron's method and simple Markov Strategies in stochastic games and control

We introduce a modification of Perron's method, where semi-solutions are considered in a carefully defined asymptotic sense. With this definition, we can show, in a rather elementary way, that in a zero-sum game or a control problem (with or without model uncertainty), the value function over all strategies coincides with the value function over Markov strategies discretized in time. Therefore, there are always discretized Markov $\varepsilon$-optimal strategies, (uniform with respect to the bounded initial condition). With a minor modification, the method produces a value and approximate saddle points for an asymmetric game of feedback strategies vs. counter-strategies.