Abstract or Introduction
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 Ito’s integral of a step process η. Ito's integral with respect to G-Brownian motion is constructed for a set of stochastic processes which are not necessarily quasi-continuous. Ito’s integral will be defined on an interval [0, τ ] where τ is a stopping time. This allows us to define Ito’s integral on a larger space. Finally, we give a detailed proof of Ito’s formula for stochastic processes.
- Quote paper
- Christian Bannasch (Author), 2017, Stochastic Calculus Under Sublinear Expectation and Volatility Uncertainty, Munich, GRIN Verlag, https://www.grin.com/document/512409
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