Numerical simulations for G-Brownian motion

This paper is concerned with numerical simulations for the GBrownian motion (defined by S. Peng in Stochastic Analysis and Applications, 2007, 541-567). By the definition of the G-normal distribution, we first show that the G-Brownian motions can be simulated by solving a certain kind of Hamilton-Jacobi-Bellman (HJB) equations. Then, some finite difference methods are designed for the corresponding HJB equations. Numerical simulation results of the G-normal distribution, the G-Brownian motion, and the corresponding quadratic variation process are provided, which characterize basic properties of the G-Brownian motion. We believe that the algorithms in this work serve as a fundamental tool for future studies, e.g., for solving stochastic differential equations (SDEs)/stochastic partial differential equations (SPDEs) driven by the G-Brownian motions.