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

Adrien Barrasso (1, 2) , Francesco Russo (2)

1 X - École polytechnique
2 UMA - Unité de Mathématiques Appliquées

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 ${\mathbb D}([0,T],E)$
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.

Mots clés: Markov processes; backward stochastic differential equation; decoupled mild solutions.; Martingale problem; pseudo-PDE

Domaines: Probabilités [math.PR]

Adrien Barrasso, Francesco Russo. BSDEs with no driving martingale, Markov processes and 
associated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples.. 2017. ⟨hal-01505974v1⟩