# Decoupled Mild Solutions of Path-Dependent PDEs and Integro PDEs Represented by BSDEs Driven by Cadlag Martingales

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## 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 *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*.

## Keywords

Decoupled mild solutions Martingale problem Cadlag martingale Path-dependent PDEs Backward stochastic differential equation Identification problem## Preview

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## Notes

### Acknowledgments

The authors are grateful to the anonymous Referee and Associated Editor for their stimulating comments on the first version of the paper. The research of the first named author was provided by a PhD fellowship (AMX) of the Ecole Polytechnique. The contribution of the second named author was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.

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