G-Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty

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.

Lecture notes, 114 pages

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• A theoretical framework for the pricing of contingent claims in the presence of model uncertainty
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• Stochastic maximum principle for stochastic recursive optimal control problem under volatility ambiguity
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• The Numerical Properties of G-heat equation and Related Application
• Robust valuation and risk measurement under model uncertainty
• Doob's optional sampling and maximal inequality for $G$-martingales
• A weighted central limit theorem under sublinear expectations