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: \[Y_t=ξ-\int_{t\wedge τ}^τY_r|Y_r|^q dr-\int_{t\wedge τ}^τZ_r dB_r,\qquad t\geq 0,\] where $\tau$ is a stopping time, $q$ is a positive constant and $\xi$ is a ${\mathcal{F}}_{\tau }$-measurable random variable such that $\mathbf{P}(\xi =+\infty )>0$. We study the link between these BSDE and the Dirichlet problem on a domain $D\subset {\mathbb{R}}^d$ and with boundary condition $g$, with $g=+\infty$ on a set of positive Lebesgue measure. We also extend our results for more general BSDE.

Published at http://dx.doi.org/10.1214/009117906000000746 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)