Time discretization of BSDEs with singular terminal condition using asymptotic expansion

TL;DR

This paper develops a numerical approximation method for BSDEs with singular terminal conditions by extending asymptotic expansions, enabling better near-terminal-time solutions and analyzing the error of an implicit scheme.

Contribution

It introduces an extended asymptotic expansion approach for general generators in BSDEs with singular terminal conditions, improving approximation accuracy.

Findings

The asymptotic expansion effectively approximates BSDE solutions near terminal time.

Error analysis of the backward Euler scheme shows dependence on terminal condition.

Method extends power case results to more general generator functions.

Abstract

We consider a class of backward stochastic differential equations (BSDEs) with singular terminal condition and develop a numerical scheme to approximate their solution. To this end, we extend an asymptotic development of the BSDE solution known from the power case, which arises from optimal liquidation problems, to more general generators. This expansion allows to obtain a suitable approximation of the BSDE solution close to the terminal time. Using this as a terminal condition, we analyze the error of a backward Euler implicit scheme and detail its dependence on the terminal condition.