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

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

2007-07-30

arxiv.org/abs/0707.4387v1

Annals of Probability 2007, Vol. 35, No. 3, 1071-1117

Mathematics

Probability

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

Scientific paper

10.1214/009117906000000746

In this paper we are concerned with one-dimensional backward stochastic differential equations (BSDE in short) of the following type: \[Y_t=\xi -\int_{t\wedge \tau}^{\tau}Y_r|Y_r|^q dr-\int_{t\wedge \tau}^{\tau}Z_r dB_r,\qquad t\geq 0,\] 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.

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