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

TL;DR

This paper investigates one-dimensional backward stochastic differential equations with random stopping times and singular terminal conditions, exploring their connection to Dirichlet problems with infinite boundary data and extending to more general BSDEs.

Contribution

It establishes a link between BSDEs with singular terminal conditions and Dirichlet problems with infinite boundary data, extending the theory to more general BSDEs.

Findings

Established correspondence between BSDEs with singular terminal conditions and Dirichlet problems.

Extended results to more general classes of BSDEs.

Analyzed the impact of random stopping times on solution behavior.

Abstract

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.