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 (possibly Integro) PDEs, 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 afore mentioned BSDEs. This concept generalizes a similar notion introduced by the authors in previous 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 (in our language) 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 $u$.
Type de document :
Pré-publication, Document de travail
2018
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Contributeur : Francesco Russo <>
Soumis le : mardi 24 avril 2018 - 06:01:03
Dernière modification le : jeudi 17 mai 2018 - 09:29:41
Document(s) archivé(s) le : mercredi 19 septembre 2018 - 00:59:49

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  • HAL Id : hal-01774823, version 1
  • ARXIV : 1804.08903

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Adrien Barrasso, Francesco Russo. Decoupled mild solutions of path-dependent PDEs and IPDEs represented by BSDEs driven by cadlag martingales.. 2018. 〈hal-01774823〉

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