# Backward Stochastic Differential Equations with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples.

may, 2021
Publication type:
Paper in peer-reviewed journals
Journal:
Journal of Theoretical Probabililty.
HAL:
arXiv:
Keywords :
Martingale problem; pseudo-PDE; Markov processes; backward stochastic differential equation; decoupled mild solutions.
Abstract:
Let $(\mathbb{P}^{s,x})_{(s,x)\in[0,T]\times E}$ be a family of probability measures,  where $E$ is a Polish space, defined on the canonical probability space ${\mathbbm D}([0,T],E)$ of $E$-valued cadlag functions. We suppose that a martingale problem with  respect to a time-inhomogeneous generator $a$ is well-posed. We consider also an associated semilinear {\it Pseudo-PDE} % with generator $a$ for which we introduce a notion of so called {\it decoupled mild} solution  and study the equivalence with the notion of martingale solution introduced in a companion paper. We also investigate well-posedness for decoupled mild solutions and their relations with a special class of BSDEs without driving martingale. The notion of decoupled mild solution is a good candidate to replace the notion of viscosity solution which is not always suitable when the map $a$ is not a PDE operator.
BibTeX:
@article{Bar-Rus-2021-1,
author={Adrien Barrasso and Francesco Russo },
title={Backward Stochastic Differential Equations with no driving
martingale, Markov processes and associated Pseudo Partial
Differential Equations. Part II: Decoupled mild solutions and
Examples. },
doi={10.1007/s10959-021-01092-7 },
journal={Journal of Theoretical Probabililty. },
year={2021 },
month={5},
}