A Stochastic Game Theoretic Differential Dynamic Programming (SGT-DDP) algorithm is derived to solve a differential game under stochastic dynamics. We present the update law for the minimizing and maximizing controls for both players and provide a set of backward differential equations for the second order value function approximation. We compute the extra terms in the backward propagation equations that arise from the stochastic assumption compared with the original GTDDP. We present the SGT-DDP algorithm and analyze how the design of the cost function affects the feed-forward and feedback parts of the control policies under the game theoretic formulation. The performance of SGT-DDP is then investigated through simulations on two examples, namely, a first order nonlinear system, the inverted pendulum and the cart pole problems with conflicting controls. We conclude with some possible future extensions.