This paper introduces a novel approach for designing optimal control using data-driven Stochastic Game Theoretic Differential Dynamic Programming (SGT-DDP). The proposed method addresses unknown stochastic systems by approximating both drift and diffusion dynamics. The drift dynamics is estimated via Gaussian Process Regression (GPR) using input-output data. The diffusion dynamics is approximated from the noise data, which is extracted through subtracting the noisy output from the smoothed output. Subsequently, the binning method is combined with GPR to obtain the approximate model of the diffusion dynamics. These approximations are integrated into the SGT-DDP framework to compute optimal control policies. Simulations on benchmark nonlinear systems under unknown dynamics demonstrate the effectiveness of the method.