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 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.
