We focus on a class of path-dependent problems which include path-dependent PDEs and Integro PDEs (in short IPDEs), and their representation via BSDEs driven by a cadlag martingale. For those equations we introduce the notion of decoupled mild solution for which, under general assumptions, we study existence and uniqueness and its representation via the aforementioned BSDEs. This concept generalizes a similar notion introduced by the authors in recent papers in the framework of classical PDEs and IPDEs. For every initial condition (s, η), where s is an initial time and η an initial path, the solution of such BSDE produces a couple of processes (Y^{s, η}, Z^{s, η}). In the classical (Markovian or not) literature the function \(u(s,\eta ):= Y^{s,\eta }_{s}\) constitutes a viscosity type solution of an associated PDE (resp. IPDE); our approach allows not only to identify u as the unique decoupled mild solution, but also to solve quite generally the so called identification problem, i.e. to also characterize the (Z^{s, η})_{s, η} processes in term of a deterministic function v associated to the (above decoupled mild) solution u.

中文翻译：

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由Cadlag Martingales驱动的BSDE代表的路径相关PDE和Integro PDE的解耦温和解

我们关注一类与路径有关的问题，包括与路径有关的PDE和Integro PDE（简称IPDE），以及它们通过由族驱动的BSDE表示。对于这些方程式，我们引入去耦温和解的概念，在一般假设下，我们通过前述的BSDE研究存在和唯一性及其表示形式。这个概念概括了作者在最近的论文中在经典PDE和IPDE框架中引入的类似概念。对于每个初始条件（s，η），其中s是初始时间，η是初始路径，这种BSDE的解会产生几个过程（Y ^s，η，Z ^s，η）。在经典的（无论是否为马尔可夫的）文献中，函数\（u（s，\ eta）：= Y ^ {s，\ eta} _ {s} \）构成关联的PDE（分别为IPDE）的粘度类型解。 ; 我们的方法不仅允许将u识别为唯一的解耦温和解，而且还可以解决一般性的所谓识别问题，即也可以根据确定性函数v关联表征（Z ^s，η）_s，η过程到（上面解耦的温和的）解u。