Growth condition on the generator of BSDE with singular terminal value ensuring continuity up to terminal time

Abstract

We study the limit behavior of the solution of a backward stochastic differential equation when the terminal condition is singular, that is it can be equal to infinity with a positive probability. In the Markovian setting, Malliavin’s calculus enables us to prove continuity if a balance condition between the growth w.r.t. $y$ and the growth w.r.t. $z$ of the generator is satisfied. As far as we know, this condition is new. We apply our result to liquidity problem in finance and to the solution of some semi-linear partial differential equation ; the imposed assumption is also new in the literature on PDE.