Zero-sum games for Volterra integral equations and viscosity solutions of path-dependent Hamilton–Jacobi equations

¹ N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia.
² Ural Federal University, Ekaterinburg, Russia.

^* Corresponding author: m.i.gomoyunov@gmail.com

Received: 18 April 2024
Accepted: 24 April 2025

Abstract

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.