Discrete-time approximation of multidimensional BSDEs with oblique reflections

TL;DR

This paper develops a natural discrete-time approximation scheme for multidimensional reflected backward stochastic differential equations (BSDEs) with oblique reflections, providing error bounds and analyzing convergence properties.

Contribution

It introduces a new approximation scheme based on oblique projections for multidimensional reflected BSDEs, improving upon previous penalization methods.

Findings

Error on grid points is of order 1/2 minus epsilon when driver is independent of Z

The scheme effectively controls the approximation error for multidimensional reflected BSDEs

Provides theoretical error bounds and convergence analysis for the proposed method

Abstract

In this paper, we study the discrete-time approximation of multidimensional reflected BSDEs of the type of those presented by Hu and Tang [Probab. Theory Related Fields 147 (2010) 89-121] and generalized by Hamad\`ene and Zhang [Stochastic Process. Appl. 120 (2010) 403-426]. In comparison to the penalizing approach followed by Hamad\`{e}ne and Jeanblanc [Math. Oper. Res. 32 (2007) 182-192] or Elie and Kharroubi [Statist. Probab. Lett. 80 (2010) 1388-1396], we study a more natural scheme based on oblique projections. We provide a control on the error of the algorithm by introducing and studying the notion of multidimensional discretely reflected BSDE. In the particular case where the driver does not depend on the variable $Z$, the error on the grid points is of order $1/2-\varepsilon$, $\varepsilon>0$.