Backward stochastic differential equations with random stopping time and singular final condition

Abstract

In this paper we are concerned with one-dimensional backward stochastic differential equations (BSDE in short) of the following type:

Y_t=ξ−∫_t∧τ^τY_r|Y_r|^q dr−∫_t∧τ^τZ_r dB_r, t≥0,

where τ is a stopping time, q is a positive constant and ξ is a ℱ_τ-measurable random variable such that P(ξ=+∞)>0. We study the link between these BSDE and the Dirichlet problem on a domain D⊂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.