#### Degree

Master of Science

#### Program

Statistics and Actuarial Sciences

Kulperger, Reg

#### Abstract

The $G$-expectation framework, motivated by problems with \emph{uncertainty}, is a new generalization of the classical probability framework. Similar to the Choquet expectation, the $G$-expectation can be represented as the supremum of a class of linear expectations. In the past two decades, it has developed into a complete stochastic structure connected with a large family of nonlinear PDEs. Nonetheless, to apply it to real-world problems with uncertainty, it is fundamentally necessary to build up the associated statistical methodology.

This thesis explores the \emph{computation, simulation, and estimation} of the $G$-normal distribution (a typical distribution with variance uncertainty) by constructing a new substructure called the \emph{Semi-$G$-normal distribution} which provides the transition from classical normal to $G$-normal distribution. Interestingly, it also gives an efficient iterative scheme to stochastically solve the nonlinear \emph{Black-Scholes-Barenblatt equation with volatility uncertainty}. This thesis is the theoretical and technical preparation for the future industrial application of $G$-framework.

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