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We study a class of nonlinear BSDEs with a superlinear driver process fadapted to a filtration F and over a random time interval [[0, S]] where S is astopping time of F. The terminal condition $\xi$ is allowed to take the value+$\infty$, i.e., singular. Our goal is to show existence of solutions to theBSDE in this setting. We will do so by proving that the minimal supersolutionto the BSDE is a solution, i.e., attains the terminal values with probability1. We consider three types of terminal values: 1) Markovian: i.e., $\xi$ is ofthe form $\xi$ = g($\Xi$ S) where $\Xi$ is a continuous Markovian diffusionprocess and S is a hitting time of $\Xi$ and g is a deterministic function 2)terminal conditions of the form $\xi$ = $\infty$ $\times$ 1 {$\tau$ $\le$S} and3) $\xi$ 2 = $\infty$ $\times$ 1 {$\tau$ >S} where $\tau$ is another stoppingtime. For general $\xi$ we prove the minimal supersolution is continuous attime S provided that F is left continuous at time S. We call a stopping time Ssolvable with respect to a given BSDE and filtration if the BSDE has a minimalsupersolution with terminal value $\infty$ at terminal time S. The concept ofsolvability plays a key role in many of the arguments. Finally, we discussimplications of our results on the Markovian terminal conditions to solution ofnonlinear elliptic PDE with singular boundary conditions.

Continuity problem for singular BSDE with random terminal time