Limit behaviour of BSDE with jumps and with singular terminal condition

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.