# Decoupled mild solutions of path-dependent PDEs and IPDEs represented by BSDEs driven by cadlag martingales.

Abstract : 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 {\it 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 $(s,\eta)$, where $s$ is an initial time and $\eta$ an initial path, the solution of such BSDE produces a couple of processes $(Y^{s,\eta},Z^{s,\eta})$. 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 {\it identification problem}, i.e. to also characterize the $(Z^{s,\eta})_{s,\eta}$ processes in term of a deterministic function $v$ associated to the (above decoupled mild) solution $u$.
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Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-01774823
Contributor : Francesco Russo <>
Submitted on : Friday, March 8, 2019 - 7:10:44 PM
Last modification on : Friday, April 12, 2019 - 1:31:47 AM

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### Identifiers

• HAL Id : hal-01774823, version 2
• ARXIV : 1804.08903

### Citation

Adrien Barrasso, Francesco Russo. Decoupled mild solutions of path-dependent PDEs and IPDEs represented by BSDEs driven by cadlag martingales.. 2019. ⟨hal-01774823v2⟩

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