G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty

TL;DR

This paper develops a new framework for stochastic calculus and risk analysis under volatility uncertainty using G-normal distributions and G-Brownian motion, extending classical probability concepts to sublinear expectation spaces.

Contribution

It introduces G-normal distributions and G-Brownian motion, establishing a new stochastic calculus framework and statistical methods for risk analysis under volatility uncertainty.

Findings

Established G-normal distributions and G-Brownian motion.

Developed stochastic calculus tools like Ito's integral and formula.

Proved law of large numbers and central limit theorem in sublinear expectation context.

Abstract

We introduce a new notion of G-normal distributions. This will bring us to a new framework of stochastic calculus of Ito's type (Ito's integral, Ito's formula, Ito's equation) through the corresponding G-Brownian motion. We will also present analytical calculations and some new statistical methods with application to risk analysis in finance under volatility uncertainty. Our basic point of view is: sublinear expectation theory is very like its special situation of linear expectation in the classical probability theory. Under a sublinear expectation space we still can introduce the notion of distributions, of random variables, as well as the notions of joint distributions, marginal distributions, etc. A particularly interesting phenomenon in sublinear situations is that a random variable Y is independent to X does not automatically implies that X is independent to Y. Two important theorems have been proved: The law of large number and the central limit theorem.