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

TL;DR

A class of numerical schemes for backward stochastic differential equation driven by $G$-Brownian motion, which is related to a fully nonlinear PDE based on Peng's central limit theorem, and it is shown that the $\theta$-scheme admits a half order convergence rate in the general case.

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 $θ$-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.

Figures (2)

• Figure 1: Convergence rates with different parameters $\theta_{i}(i=1,2)$.
• Figure 2: Convergence rates with different $\sigma^{2}$.

Theorems & Definitions (30)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Remark 2.7
• Theorem 2.8: P2010P2019
• Definition 2.9
• Definition 2.10