G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Ito's type
Abstract
We introduce a notion of nonlinear expectation --G--expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first discuss the notion of G-standard normal distribution. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a G--Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G--Brownian motion and derive the related Ito's formula. We have also give the existence and uniqueness of stochastic differential equation under our G-expectation. As compared with our previous framework of g-expectations, the theory of G-expectation is intrinsic in the sense that it is not based on a given (linear) probability space.
Keywords
Cite
@article{arxiv.math/0601035,
title = {G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Ito's type},
author = {Shige Peng},
journal= {arXiv preprint arXiv:math/0601035},
year = {2007}
}
Comments
Submited to Proceedings Abel Symposium 2005, Dedicated to Professor Kiyosi Ito for His 90th Birthday