Backward Stochastic Differential Equations with Non-Markovian Singular Terminal Conditions with General Driver and Filtration

TL;DR

This paper studies backward stochastic differential equations with complex, non-Markovian terminal conditions, proving solutions attain their terminal values and analyzing the density of exit times for diffusion processes.

Contribution

It establishes that minimal supersolutions are actual solutions under non-Markovian singular terminal conditions and proves the existence of continuous densities for certain exit times.

Findings

Minimal supersolutions are solutions, attaining their terminal values.

Exit times of diffusion processes have continuous densities.

The results apply to complex terminal conditions involving stopping times.

Abstract

We consider a class of Backward Stochastic Differential Equations with superlinear driver process $f$ adapted to a filtration supporting at least a $d$ dimensional Brownian motion and a Poisson random measure on ${\mathbb R}^m- \{0\}.$ We consider the following class of terminal conditions $\xi_1 = \infty \cdot 1_{\{\tau_1 \le T\}}$ where $\tau_1$ is any stopping time with a bounded density in a neighborhood of $T$ and $\xi_2 = \infty \cdot 1_{A_T}$ where $A_t$, $t \in [0,T]$ is a decreasing sequence of events adapted to the filtration ${\mathcal F}_t$ that is continuous in probability at $T$. A special case for $\xi_2$ is $A_T = \{\tau_2 > T\}$ where $\tau_2$ is any stopping time such that $P(\tau_2 =T) =0.$ In this setting we prove that the minimal supersolutions of the BSDE are in fact solutions, i.e., they attain almost surely their terminal values. We further show that the first exit time from a time varying domain of a $d$-dimensional diffusion process driven by the Brownian motion with strongly elliptic covariance matrix does have a continuous density; therefore such exit times can be used as $\tau_1$ and $\tau_2$ to define the terminal conditions $\xi_1$ and $\xi_2.$ The proof of existence of the density is based on the classical Green's functions for the associated PDE.