Backward stochastic differential equations with non-Markovian singular terminal conditions for general driver and filtration

We consider a class of Backward Stochastic Differential Equations with superlinear driver process f adapted to filtration supporting at least d dimensional Brownian motion and Poisson random measure on Rm∖{0}. the following terminal conditions: ξ1=∞⋅1{τ1≤T} where τ1 is any stopping time bounded density in neighborhood T ξ2=∞⋅1AT At, t∈[0,T] decreasing sequence events Ft that continuous probability (equivalently, AT={τ2>T} τ2 such P(τ2=T)=0). In this setting we prove minimal supersolutions BSDE are fact solutions, i.e., they attain almost surely their values. note first exit from varying domain d-dimensional diffusion driven by strongly elliptic covariance matrix does have density. Therefore times can be used as define conditions ξ1 ξ2. The proof existence based classical Green’s functions for associated PDE.