Decoupled mild solutions of path-dependent PDEs and IPDEsrepresented by BSDEs driven by cadlag martingales.

submitted
Publication type:
Paper in peer-reviewed journals
arXiv:
images/icons/icon_arxiv.png 1804.08903
Keywords :
Decoupled mild solutions; martingale problem; cadlag martingale; path-dependent PDEs; backward stochastic differ- ential equation; identification problem ;
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 $u$.
BibTeX:
@article{Bar-Rus-2200-2,
    author={Adrien Barrasso and Francesco Russo },
    title={Decoupled mild solutions of path-dependent PDEs and 
           IPDEsrepresented by BSDEs driven by cadlag martingales. },
    year={submitted },
    month={4},
}