Stochastic Calculus Under Sublinear Expectation and Volatility Uncertainty

Research Paper (postgraduate) from the year 2017 in the subject Mathematics - Stochastics, grade: 1,7, LMU Munich, language: English, abstract: Detailed results of stochastic calculus under probability model uncertainty have been proven by Shige Peng. At first, we give some basic properties of sublinear expectation E. One can prove that E has a representaion as the Supremum of a specific set of well known linear expectation. P is called uncertainty set and characterizes the probability model uncertainty.

Based on the results of Hu and Peng ([HP09]) we prove that P is a weakly compact set of probability measures. Based on the work of Peng et. Al. we give the definition and properties of maximal distribution and G-normal Distribution. Furthermore, G-Brownian motion and its corresponding G-expectation will be constructed. Briefly speaking, a G -Brownian motion (Bt)t¿0 is a continuous process with independent and stationary increments under a given sublinear expectation E.

In this work, we use the results in [LP11] and study Itös integral of a step process ¿. Ito's integral with respect to G-Brownian motion is constructed for a set of stochastic processes which are not necessarily quasi-continuous. Itös integral will be defined on an interval [0, ¿ ] where ¿ is a stopping time. This allows us to define Itös integral on a larger space. Finally, we give a detailed proof of Itös formula for stochastic processes.