  We study the existence of a minimal supersolution for backward stochastic
differential equations when the terminal data can take the value +$\infty$ with
positive probability. We deal with equations on a general filtered probability
space and with generators satisfying a general monotonicity assumption. With
this minimal supersolution we then solve an optimal stochastic control problem
related to portfolio liquidation problems. We generalize the existing results
in three directions: firstly there is no assumption on the underlying
filtration (except completeness and quasi-left continuity), secondly we relax
the terminal liquidation constraint and finally the time horizon can be random.
