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

The paper proves that the lower and upper value functionals for the Volterra game are viscosity solutions of a path-dependent Hamilton–Jacobi equation (Theorem 9.1), establishes a comparison/uniqueness theorem under a Lipschitz-in-gradient hypothesis (Theorem 9.2), and—under Isaacs’ condition—concludes equality of the values with uniqueness of the common viscosity solution (Theorem 9.4). These steps rely on the dynamic programming principle (Proposition 5.2) and a ci-chain-rule for ci-smooth test functionals (Proposition 8.1) on motion-invariant compact subsets Gk of the history space G, along with continuity properties of the values (Section 7). The candidate solution mirrors this architecture: it derives the viscosity inequalities for ρ− and ρ+ from DPP plus the ci-chain-rule, then invokes a comparison principle under the same Lipschitz-in-gradient structure, and finally applies Isaacs to identify the value. Minor differences are present in exposition (e.g., the candidate informally uses a short-horizon Hölder control to justify a frozen-x strategy in the supersolution half, while the paper cites a standard proof template), but the logical content and hypotheses match the paper. In particular, the paper’s Theorem 9.1 uses DPP (Prop. 5.2) and the ci-chain-rule (Prop. 8.1) in the standard way to obtain the viscosity property for ρ±, exactly as the candidate outlines . The uniqueness assumption in Theorem 9.2 (continuity and Lipschitz in s with growth c(1+||x||)) is equivalent to the candidate’s local-Lipschitz-on-Gk phrasing since x is bounded on each Gk, hence the comparison step aligns . Under Isaacs, Theorem 9.4 then yields ρ−=ρ+=ρ and uniqueness of the viscosity solution, as the candidate states . The auxiliary structural pieces (definition of the history space G, nondegeneracy Assumption 3.1, continuity on Gk, and DPP) match the candidate’s Step 1 .