Limit behaviour of BSDE with jumps and with singular terminal condition

Popier, A.

ESAIM: Probability and Statistics, Tome 20 (2016), pp. 480-509.

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 [T. Kruse and A. Popier, Stoch. Process. Appl. 126 (2016) 2554–2592], 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 right continuous with left limits 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.

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