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 m...

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Bibliographic Details
Published in:	Stochastic processes and their applications Vol. 118; no. 12; pp. 2223 - 2253
Main Author:	Peng, Shige
Format:	Journal Article
Language:	English
Published:	Amsterdam Elsevier B.V 01.12.2008 Elsevier
Series:	Stochastic Processes and their Applications
Subjects:	[formula omitted]-convexity [formula omitted]-expectation [formula omitted]-normal distribution Brownian motion BSDE Distribution theory Exact sciences and technology g-expectation G-expectation G-normal distribution BSDE SDE Nonlinear probability theory Nonlinear expectation Brownian motion Ito's stochastic calculus Ito's integral Ito's formula Gaussian process Quadratic variation process Jensen's inequality G-convexity Gaussian process Itô’s formula Itô’s integral Itô’s stochastic calculus Jensen’s inequality Mathematics Nonlinear expectation Nonlinear probability theory Probability and statistics Probability theory and stochastic processes Probability theory on algebraic and topological structures Quadratic variation process Sciences and techniques of general use SDE Stochastic analysis Stochastic processes Itô’s formula 60H10 Jensen’s inequality 60H30 g -expectation Itô’s integral BSDE Nonlinear probability theory Quadratic variation process Nonlinear expectation Brownian motion Gaussian process G -normal distribution SDE 60H05 Itô’s stochastic calculus G -convexity Heat equation Gaussian distribution Stochastic motion Stochastic process Stochastic integral Non linear equation Jensen's inequality Distribution function Convexity G-convexity Itô's stochastic calculus G-normal distribution Probability theory 50H10 G-expectation Itô's formula Itô formula Stochastic calculus Gaussian processes Non linear theory Itô's integral Expectation Application
ISSN:	0304-4149, 1879-209X
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Summary:	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 Itô’s type with respect to our G -Brownian motion, and derive the related Itô’s formula. We have also obtained the existence and uniqueness of stochastic differential equations under our G -expectation.
ISSN:	0304-4149 1879-209X
DOI:	10.1016/j.spa.2007.10.015