Backward Stochastic Differential Equations with Nonmarkovian Singular Terminal Values

TL;DR

This paper develops solutions for a class of backward stochastic differential equations with non-Markovian, singular terminal conditions, involving a heat equation with singular boundary conditions, ensuring continuous paths despite discontinuous boundary data.

Contribution

It introduces a novel approach to solving BSDEs with singular, non-Markovian terminal conditions using a related heat equation with singular boundary conditions.

Findings

Successfully solves BSDEs with singular terminal values.

Establishes continuity of solutions despite boundary discontinuities.

Connects BSDE solutions to heat equations with reaction terms and singular boundaries.

Abstract

We solve a class of BSDE with a power function $f(y) = y^q$, $q > 1$, driving its drift and with the terminal boundary condition $ \xi = \infty \cdot \mathbf{1}_{B(m,r)^c}$ (for which $q > 2$ is assumed) or $ \xi = \infty \cdot \mathbf{1}_{B(m,r)}$, where $B(m,r)$ is the ball in the path space $C([0,T])$ of the underlying Brownian motion centered at the constant function $m$ and radius $r$. The solution involves the derivation and solution of a related heat equation in which $f$ serves as a reaction term and which is accompanied by singular and discontinuous Dirichlet boundary conditions. Although the solution of the heat equation is discontinuous at the corners of the domain the BSDE has continuous sample paths with the prescribed terminal value.