标题： 鞅与通过进化半群的路径依赖偏微分方程 显示英文标题

标题： Martingales and Path-Dependent PDEs via Evolutionary Semigroups

摘要： 在本文中，我们开发了一个半群理论框架，用于对具有路径依赖终端条件的鞅进行分析表征。 我们的主要结果表明，形式为\[ V(t) - \int_0^t\Psi(s)\, ds \]的可测适应过程在期望算子$\mathbb{E}$下是鞅，当且仅当$V$的时间平移版本是涉及与$\mathbb{E}$密切相关的路径依赖微分算子的终值问题的弱解。 我们使用进化半群的概念，证明了此类具有可测终端条件的终值问题的强解和弱解的存在性和唯一性。 为了表征补偿器$\Psi$，我们引入了$V$的$\mathbb{E}$-导数概念，该概念在特殊情况下与Dupire的时间导数一致。 我们还根据Dupire导数比较了我们的结果与路径依赖偏微分方程，例如路径依赖热方程。 显示英文摘要

摘要： 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.