Zero-Sum Games for Volterra Integral Equations and Viscosity Solutions of Path-Dependent Hamilton-Jacobi Equations

We consider a game, in which the dynamics is described by a non-linear Volterra integral equation of Hammerstein type with a weakly-singular kernel and the goals of the first and second players are, respectively, to minimize and maximize a given cost functional. We propose a way of how the dynamic programming principle can be formalized and the theory of generalized (viscosity) solutions of path-dependent Hamilton--Jacobi equations can be developed in order to prove the existence of the game value, obtain a characterization of the value functional, and construct players' optimal feedback strategies.