Martingales and Path-Dependent PDEs via Evolutionary Semigroups

TL;DR

This paper introduces a semigroup-based framework to characterize martingales with path-dependent terminal conditions, connecting them to solutions of path-dependent PDEs using evolutionary semigroups and derivatives.

Contribution

It develops a novel semigroup-theoretic approach to analyze martingales and path-dependent PDEs, establishing existence, uniqueness, and characterizations via evolutionary semigroups and derivatives.

Findings

Established a characterization of martingales via path-dependent PDEs.

Proved existence and uniqueness of solutions for these PDEs.

Connected the $ ext{E}$-derivative to Dupire's derivatives in special cases.

Abstract

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.