Martingales and Path-Dependent PDEs via Evolutionary Semigroups

In this article, we develop a semigroup-theoretic framework for the analytic characterisation of martingales with path-dependent terminal conditions. Our main result establishes that a measurable adapted process of the form $$V(t) - \int_0^t\Psi(s)\, ds$$ is a martingale with respect to an expectation operator $\mathbb{E}$ 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 $\mathbb{E}$. 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 characterise the compensator $\Psi$, we introduce the notion of $\mathbb{E}$-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.