Continuity problem for singular BSDE with random ter-minal time

We study a class of nonlinear BSDEs with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition xi is allowed to take the value +infinity, i.e., singular. Our goal is to show existence of solutions to the BSDE in this setting. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., attains the terminal values with probability 1. We consider three types of terminal values: 1) Markovian: i.e., xi is of the form xi = g(Xi(S)) where Xi is a continuous Markovian diffusion process and S is a hitting time of Xi and g is a deterministic function 2) terminal conditions of the form xi(1) = infinity.1({tau <= S}) and 3) xi(2) = infinity.1({tau > S}) where tau is another stopping time. For general xi we prove the minimal supersolution is continuous at time S provided that F is left continuous at time S. We call a stopping time S solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value infinity at terminal time S. The concept of solvability plays a key role in many of the arguments. Finally, we discuss implications of our results on the Markovian terminal conditions to solution of nonlinear elliptic PDE with singular boundary conditions.