Limit behaviour of BSDE with jumps and with singular terminal condition

TL;DR

This paper investigates the behavior of minimal solutions to backward stochastic differential equations with jumps and singular terminal conditions, establishing conditions under which solutions have well-defined limits at terminal time and match the terminal data.

Contribution

It provides conditions ensuring the solution's existence, left limits at terminal time, and equality with the potentially infinite terminal condition.

Findings

Solutions have left limits at terminal time under certain conditions.

The left limit of the solution equals the terminal condition.

Existence of minimal solutions with singular terminal data is confirmed.

Abstract

We study the behaviour at the terminal time T of the minimal solution of a backward stochastic differential equation when the terminal data can take the value +$\infty$ with positive probability. In a previous paper, we have proved existence of this minimal solution (in a weak sense) in a quite general setting. But two questions arise in this context and were still open: is the solution c{\`a}d{\`i}{\`a}g on [0,T] ? In other words does the solution have a left limit at time T ? The second question is: is this limit equal to the terminal condition? In this paper, under additional conditions on the generator and the terminal condition, we give a positive answer to these two questions.