Convergence of Discrete-Time Deterministic Games to Path-Dependent Isaacs Partial Differential Equations Under Quadratic Growth Conditions

We consider discrete-time approximations for path-dependent Isaacs partial differential equations (PDEs) of deterministic differential games under quadratic growth conditions including linear/quadratic problems with distributed and discrete delays. Owing to the path-dependence of the system, the Isaacs PDEs are defined on infinite-dimensional spaces of past state trajectories. Using the notion of viscosity solutions on the infinite-dimensional spaces as proposed by Lukoyanov based on co-invariant derivatives of path spaces, we show that the discrete-time path-dependent dynamic games converge to a unique viscosity solution for the Isaacs PDEs. Noting that these games can be practically defined on finite-dimensional spaces, we discuss finite-dimensional approximations of viscosity solutions of path-dependent Isaacs PDEs. Given an example, we derive discrete-time Riccati-type recursive equations to calculate explicit discrete-time approximations for the path-dependent linear/quadratic problems.