Martingales and path-dependent PDEs via evolutionary semigroups

In this article, we develop a semigroup-theoretic framework for the analytic characterization of martingales with path-dependent terminal conditions. Our main result establishes that a measurable adapted process of the form V(t) - ∫ ₀ᵗ Ψ (s)\, ds V ( t ) - ∫ 0 t Ψ ( s ) d s is a martingale with respect to an expectation operator E E if and only if a time-shifted version of V is a mild solution of a final value problem involving a path-dependent differential operator that is intrinsically connected to E E . We prove existence and uniqueness of strong and mild solutions for such final value problems with measurable terminal conditions using the concept of evolutionary semigroups. To characterize the compensator Ψ Ψ , we introduce the notion of the E E -derivative of V , which in special cases coincides with Dupire’s time derivative. We also compare our findings to path-dependent partial differential equations in terms of Dupire derivatives such as the path-dependent heat equation.