Backward Stochastic Differential Equations with No Driving Martingale, Markov Processes and Associated Pseudo-Partial Differential Equations: Part II—Decoupled Mild Solutions and Examples

Let $$(\mathbb {P}^{s,x})_{(s,x)\in [0,T]\times E}$$ ( P s , x ) ( s , x ) ∈ [ 0 , T ] × E be a family of probability measures, where E is a Polish space, defined on the canonical probability space $${\mathbb D}([0,T],E)$$ D ( [ 0 , T ] , E ) of E-valued càdlàg functions. We suppose that a martingale problem with respect to a time-inhomogeneous generator a is well-posed. We consider also an associated semilinear Pseudo-PDE for which we introduce a notion of so-called 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 backward stochastic differential equations (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.