Backward stochastic differential equations with non-Markovian singular terminal values

Abstract

We solve a class of BSDE with a power function [Formula: see text], [Formula: see text], driving its drift and with the terminal boundary condition [Formula: see text] (for which [Formula: see text] is assumed) or [Formula: see text], where [Formula: see text] is the ball in the path space [Formula: see text] of the underlying Brownian motion centered at the constant function [Formula: see text] and radius [Formula: see text]. The solution involves the derivation and solution of a related heat equation in which [Formula: see text] 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.