  This paper presents a novel approach to numerically solve stochastic
differential games for nonlinear systems. The proposed approach relies on the
nonlinear Feynman-Kac theorem that establishes a connection between parabolic
deterministic partial differential equations and forward-backward stochastic
differential equations. Using this theorem the Hamilton-Jacobi-Isaacs partial
differential equation associated with differential games is represented by a
system of forward-backward stochastic differential equations. Numerical
solution of the aforementioned system of stochastic differential equations is
performed using importance sampling and a Long-Short Term Memory recurrent
neural network, which is trained in an offline fashion. The resulting algorithm
is tested on two example systems in simulation and compared against the
standard risk neutral stochastic optimal control formulations.
