140 research outputs found
Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation
We develop a notion of nonlinear expectation --G-expectation-- generated by a
nonlinear heat equation with infinitesimal generator G. We first study
multi-dimensional G-normal distributions. With this nonlinear distribution we
can introduce our G-expectation under which the canonical process is a multi
dimensional 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 obtained
the existence and uniqueness of stochastic differential equation under our
G-expectation.Comment: 27 page
G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Ito's type
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.Comment: Submited to Proceedings Abel Symposium 2005, Dedicated to Professor
Kiyosi Ito for His 90th Birthda
- …