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

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.