Decoupled mild solutions of path-dependent PDEs and Integro PDEs represented by BSDEs driven by cadlag martingales
july, 2020
Publication type:
Paper in peer-reviewed journals
Journal:
Potential Analysis., vol. 53, pp. 449-481
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Keywords :
Decoupled mild solutions; martingale problem; cadlag martingale; path-dependent PDEs; backward stochastic differential equation; identification problem.
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 $u$.
BibTeX:
@article{Bar-Rus-2020, author={Adrien Barrasso and Francesco Russo }, title={Decoupled mild solutions of path-dependent PDEs and Integro PDEs represented by BSDEs driven by cadlag martingales }, doi={10.1007/s11118-019-09775-x }, journal={Potential Analysis. }, year={2020 }, month={7}, volume={53 }, pages={449--481}, }