Backward Stochastic Differential Equations with Random Stopping Time and Singular Final Condition

In this paper we are concerned with one-dimensional backward stochastic differential equations (BSDE in short) of the following type: $Y_{t}=\xi -\int_{t\wedge \tau}^{\tau }Y_{r}|Y_{r}|^{q}\ dr-\int_{t\wedge \tau}^{\tau }Z_{r}\ dB_{r},\quad \quad t\geq 0$, where τ is a stopping time, q is a positive constant and ξ is a $\scr{F}_{\tau}\text{-measurable}$ random variable such that P(ξ = +∞) > 0. We study the link between these BSDE and the Dirichlet problem on a domain $D\subset {\Bbb R}^{d}$ and with boundary condition g, with g = +∞ on a set of positive Lebesgue measure. We also extend our results for more general BSDE.