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In this paper, we consider state and control path-dependent stochasticzero-sum differential games, where the dynamics and the running cost includeboth state and control paths of the players. Using the notion ofnonanticipative strategies, we define lower and upper value functionals, whichare functions of the initial state and control paths of the players. We provethat the value functionals satisfy the dynamic programming principle. Theassociated lower and upper Hamilton-Jacobi-Isaacs (HJI) equations from thedynamic programming principle are state and control path-dependent nonlinearsecond-order partial differential equations. We apply the functional It\^ocalculus to prove that the lower and upper value functionals are viscositysolutions of (lower and upper) state and control path-dependent HJI equations,where the notion of viscosity solutions is defined on a compact subset of an$\kappa$-H\"older space introduced in \cite{Tang_DCD_2015}. Moreover, we showthat the Isaacs condition and the uniqueness of viscosity solutions imply theexistence of the game value. For the state path-dependent case, we prove theuniqueness of classical solutions for the (state path-dependent) HJI equations.

State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton-Jacobi-Isaacs Equations