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We develop here the Stochastic Perron Method in the framework of two-playerzero-sum differential games. We consider the formulation of the game where bothplayers play, symmetrically, feed-back strategies (as in [CR09] or [PZ12]) asopposed to the Elliott-Kalton formulation prevalent in the literature. Theclass of feed-back strategies we use is carefully chosen so that the stateequation admits strong solutions and the technicalities involved in theStochastic Perron Method carry through in a rather simple way. More precisely,we define the game over elementary strategies, which are well motivated byintuition. Within this framework, the Stochastic Perron Method produces aviscosity sub-solution of the upper Isaacs equation dominating the upper valueof the game, and a viscosity super-solution of the upper Isaacs equation lyingbelow the upper value of the game. Using a viscosity comparison result weobtain that the upper value is the unique and continuous viscosity solution ofthe upper Isaacs equation. An identical statement holds for the lower value andlower Isaacs equation. A version of the Dynamic Programming Principle isobtained as a by-product. If the Isaacs condition holds, the game has a valueover elementary (pure) strategies.

Stochastic Perron's method and elementary strategies for zero-sum differential games