The Numerical Simulation Of G-Brownian Motion

The aim of this paper is to numerically simulate the G-normal distribution and G-Brownian motion as well as analyze the accuracy and effectiveness of the numerical methods.The sublinear expectation is a basic tool with application to risk analysis in finance under volatility uncertainty. In this new framework, there is a notion of G-normal distribution corresponding to the well-known normal distribution in classical probability theory. And there are in-depth theoretical study for G-Brownian motion, stochastic calculus of Ito’s type, stochastic differential equations corresponding to the G-Brownian motion. Indeed, G-Brownian mo-tion is of great significance in theory and practice.We study the G-normal distribution and G-Brownian motion in the per-spective of numerical simulation. G-normal distribution M(0;[σ2, σ2]) is u-niquely defined by its G-heat equation (?)u/(?)t-G((?)2u/(?)x2)=0,u(0,x)=φ(x), where G(a):=1/2E[X2a]=1/2(σ2a+-σ2a-),a∈R, a+=max{0,a}, a-=(-a)+. So, we start with the numerical methods to solve the G-heat equation. We use four schemes:explicit difference scheme, semi-implicit difference scheme, implicit difference scheme and Crank-Nicolson scheme to solve it, and we give the method to simulate the G-normal distribution. Then we get the simula-tion method of G-Brownian motion by the relationship of G-normal distribu-tion and G-Brownian motion, and give the simulation results of G-Brownian motion using some pictures. We also analyze the relationship between the simulation trajectory of G-Brownian motion and σ2,σ2. Moreover, we give two examples to demonstrate the accuracy and effectiveness of the numeri-cal methods. These simulation results will be helpful to the understanding of G-normal distribution and G-Brownian motion, and will drive their study in theory and practice.