Continuity problem for singular BSDE with random terminal time

TL;DR

This paper investigates the existence of solutions to a class of nonlinear backward stochastic differential equations with singular terminal conditions and random terminal times, extending the theory to include infinite terminal values and implications for PDEs.

Contribution

It introduces the concept of solvability for BSDEs with singular terminal conditions and proves the existence of solutions in this challenging setting, including Markovian and non-Markovian cases.

Findings

Minimal supersolutions attain the terminal value with probability 1.

Solutions are continuous at the terminal time under certain conditions.

Implications for solving nonlinear elliptic PDEs with singular boundary conditions.

Abstract

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 +$\infty$, 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 $\xi$ = $\infty$ $\times$ 1 {$\tau$ $\le$S} and 3) $\xi$ 2 = $\infty$ $\times$ 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 $\infty$ 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.