Stochastic Integrals With Respect To The Symmetric G-martingales And The Law Of Large Numbers Under Nonlinear Expectation

Motivated by uncertainty problems in Finance and Economics, Peng has in-troduced recently a new notion of nonlinear expectation, the so-called G-expectation. Together with the notion of G-expectation Peng also introduced the G-normal distribution and G-martingales. Moreover, Peng developed an Ito calculus for the G-Brownian motion. Recently, Xu Jing, Qian Lin and Zhang Bo study the stochastic calculus with respect to symmetric G-martingales to a large class of G-martingales and study related properties. Our paper organized as follows. Firstly, we obtain the G-Ito formula with respect to symmetric G-martingales integral. Secondly, we study the Stratonovich integral of symmetric G-martingales.Thirdly, we drive the concept of strong law of large numbers and weak law of large num-bers. Finally, we obtain two strong limit theorems of large numbers under certain condition.