We consider a class of Backward Stochastic Differential Equations with superlinear driver process f adapted to a filtration supporting at least a d dimensional Brownian motion and a Poisson random measure on . We consider the following class of terminal conditions: where is any stopping time with a bounded density in a neighborhood of T and where , is a decreasing sequence of events adapted to the filtration that is continuous in probability at T (equivalently, where is any stopping time such that ). In this setting we prove that the minimal supersolutions of the BSDE are in fact solutions, i.e., they attain almost surely their terminal values. We note that the first exit time from a time varying domain of a d-dimensional diffusion process driven by the Brownian motion with strongly elliptic covariance matrix does have a continuous density. Therefore such exit times can be used as and to define the terminal conditions and . The proof of existence of the density is based on the classical Green’s functions for the associated PDE.
This work is supported by TUBITAK (The Scientific and Technological Research Council of Turkey) through project number 118F163. We are grateful for this support.
"Backward stochastic differential equations with non-Markovian singular terminal conditions for general driver and filtration." Electron. J. Probab. 26 1 - 27, 2021. https://doi.org/10.1214/21-EJP619