Backward stochastic differential equations with singular terminal condition

In this paper, we are concerned with backward stochastic differential equations (BSDE for short) of the following type: Y t = ξ − ∫ t T Y r | Y r | q d r − ∫ t T Z r d B r , where q is a positive constant and ξ is a random variable such that P ( ξ = + ∞ ) > 0 . We study the link between these BSDE and the associated Cauchy problem with terminal data g , where g = + ∞ on a set of positive Lebesgue measure.