G-Expectation And Its Related Calculation Problems

When applying mathematics theory into the financial world, a key problem is to price the risk assets and their derivatives. In the financial world, we often use linear probability theory to depict the random process, but in 1952, Allais Paradox which showed the shortage of linear probability theory was put forward. After that, mathematicians and financial experts started to study nonlinear expectation. Shige Peng introduced the concept of G-Expectation which is generated by a parabolic PDE, as a sub-linear expectation, G-Expectation has a wide range of applications in financial world.In this thesis, we mainly focus on G-Expectations and its related calculation problems. The main results we have got are as follow: 1. In the classical linear probability theory frame, we verified the continuity of the solution of G-heat equation (2-2) at the point of t =0,after that, We proved the equivalence of the two different definitions of G-Normal Distribution; 2. Using the comparison theorem of ODE and the Weierstrass approximation theorem, we obtained some estimators of E [ ( X)]; 3. From the perspective of the G-heat equation (2-4), we gave an upper bound estimate under certain condition, and discussed the general method to solve the G-heat equation (2-4).