Convergence to a self-normalized G-Brownian motion

Zhengyan Lin; Li-Xin Zhang

doi:10.1186/s41546-017-0013-8

Article Contents

2017, Volume 2: 4. Doi: 10.1186/s41546-017-0013-8

This volume Previous Article Backward stochastic differential equations with Young drift Next Article A survey of time consistency of dynamic risk measures and dynamic performance measures in discrete time: LM-measure perspective

Convergence to a self-normalized G-Brownian motion

Zhengyan Lin and
Li-Xin Zhang^,

School of Mathematical Sciences, Zhejiang University, Hangzhou 310027 China

Li-Xin Zhang, stazlx@zju.edu.cn

Received: October 24, 2016

Revised: January 04, 2017

Published: September 2020

supported by Grants from the National Natural Science Foundation of China (No. 11225104), the 973 Program (No. 2015CB352302) and the Fundamental Research Funds for the Central Universities.

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Abstract

G-Brownian motion has a very rich and interesting new structure that nontrivially generalizes the classical Brownian motion. Its quadratic variation process is also a continuous process with independent and stationary increments. We prove a self-normalized functional central limit theorem for independent and identically distributed random variables under the sub-linear expectation with the limit process being a G-Brownian motion self-normalized by its quadratic variation. To prove the self-normalized central limit theorem, we also establish a new Donsker's invariance principle with the limit process being a generalized G-Brownian motion.

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Mathematics Subject Classification: 60F15;60F05;60H10;60G48.

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