Discrete-time approximation for backward stochastic differential equations driven by $G$-Brownian motion

TL;DR

This paper develops and analyzes discrete-time schemes for approximating backward stochastic differential equations driven by $G$-Brownian motion, providing convergence rates and numerical methods for applications in financial hedging.

Contribution

Introduces $ heta$-schemes for $G$-BSDEs using an extended $ ilde{G}$-expectation space, achieving half-order convergence and sometimes first-order, with practical numerical implementation.

Findings

Schemes achieve half-order convergence in general cases.

Special cases attain first-order convergence.

Numerical examples confirm theoretical convergence rates.

Abstract

In this paper, we study the discrete-time approximation schemes for a class of backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs) which corresponds to the hedging pricing of European contingent claims. By introducing an auxiliary extended $\widetilde{G}$-expectation space, we propose a class of $\theta$-schemes to discrete $G$-BSDEs in this space. With the help of nonlinear stochastic analysis techniques and numerical analysis tools, we prove that our schemes admit half-order convergence for approximating $G$-BSDE in the general case. In some special cases, our schemes can achieve a first-order convergence rate. Finally, we give an implementable numerical scheme for $G$-BSDEs based on Peng's central limit theorem and illustrate our convergence results with numerical examples.