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., vol. 34, pp. 1110-1148
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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}, volume={34 }, pages={1110--1148}, }