BSDEs with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples.
Résumé
Let (Ps,x)(s,x)∈[0,T]×E be a family of probability measures, where E is a Polish space,defined on the canonical probability space 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.
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