LÉVY AREA ANALYSIS AND PARAMETER ESTIMATION FOR FOU PROCESSES VIA NON-GEOMETRIC ROUGH PATH THEORY*

doi:10.1007/s10473-024-0501-8

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (5): 1609-1638.doi: 10.1007/s10473-024-0501-8

• • 下一篇

LÉVY AREA ANALYSIS AND PARAMETER ESTIMATION FOR FOU PROCESSES VIA NON-GEOMETRIC ROUGH PATH THEORY^*

Zhongmin Qian¹, Xingcheng Xu^2,†

1. Mathematical Institute, University of Oxford, OX2 6GG, United Kingdom;
2. Shanghai Artificial Intelligence Laboratory, Shanghai 200232, China;
3. School of Mathematical Sciences, Peking University, Beijing 100871, China

收稿日期:2023-03-04 修回日期:2024-04-29 出版日期:2024-10-25 发布日期:2024-10-22
通讯作者: †Xingcheng X,E-mail,: xingcheng.xu18@gmail.com
作者简介:Zhongmin Qian , E-mail : qianz@maths.ox.ac.uk
基金资助:
Xu's research was supported by Shanghai Artiflcial Intelligence Laboratory.

LÉVY AREA ANALYSIS AND PARAMETER ESTIMATION FOR FOU PROCESSES VIA NON-GEOMETRIC ROUGH PATH THEORY^*

Zhongmin Qian¹, Xingcheng Xu^2,†

1. Mathematical Institute, University of Oxford, OX2 6GG, United Kingdom;
2. Shanghai Artificial Intelligence Laboratory, Shanghai 200232, China;
3. School of Mathematical Sciences, Peking University, Beijing 100871, China

Received:2023-03-04 Revised:2024-04-29 Online:2024-10-25 Published:2024-10-22
Contact: †Xingcheng X,E-mail,: xingcheng.xu18@gmail.com
About author:Zhongmin Qian , E-mail : qianz@maths.ox.ac.uk
Supported by:
Xu's research was supported by Shanghai Artiflcial Intelligence Laboratory.

摘要/Abstract

摘要： This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting. To tackle this problem, we propose a novel approach based on rough path theory that allows us to construct pathwise rough path estimators from both continuous and discrete observations of a single path. Our approach is particularly suitable for high-frequency data. To formulate the parameter estimators, we introduce a theory of pathwise Itô integrals with respect to fractional Brownian motion. By establishing the regularity of fractional Ornstein-Uhlenbeck processes and analyzing the long-term behavior of the associated Lévy area processes, we demonstrate that our estimators are strongly consistent and pathwise stable. Our findings offer a new perspective on estimating the drift parameter matrix for fractional Ornstein-Uhlenbeck processes in multi-dimensional settings, and may have practical implications for fields including finance, economics, and engineering.

关键词: Itô integration, Lévy area, non-geometric rough path, fOU processes, pathwise stability, long time asymptotic, high-frequency data

Abstract: This paper addresses the estimation problem of an unknown drift parameter matrix for a fractional Ornstein-Uhlenbeck process in a multi-dimensional setting. To tackle this problem, we propose a novel approach based on rough path theory that allows us to construct pathwise rough path estimators from both continuous and discrete observations of a single path. Our approach is particularly suitable for high-frequency data. To formulate the parameter estimators, we introduce a theory of pathwise Itô integrals with respect to fractional Brownian motion. By establishing the regularity of fractional Ornstein-Uhlenbeck processes and analyzing the long-term behavior of the associated Lévy area processes, we demonstrate that our estimators are strongly consistent and pathwise stable. Our findings offer a new perspective on estimating the drift parameter matrix for fractional Ornstein-Uhlenbeck processes in multi-dimensional settings, and may have practical implications for fields including finance, economics, and engineering.

Key words: Itô integration, Lévy area, non-geometric rough path, fOU processes, pathwise stability, long time asymptotic, high-frequency data

中图分类号:

60H05

Zhongmin Qian, Xingcheng Xu. LÉVY AREA ANALYSIS AND PARAMETER ESTIMATION FOR FOU PROCESSES VIA NON-GEOMETRIC ROUGH PATH THEORY^*[J]. 数学物理学报(英文版), 2024, 44(5): 1609-1638.

Zhongmin Qian, Xingcheng Xu. LÉVY AREA ANALYSIS AND PARAMETER ESTIMATION FOR FOU PROCESSES VIA NON-GEOMETRIC ROUGH PATH THEORY^*[J]. Acta mathematica scientia,Series B, 2024, 44(5): 1609-1638.

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LÉVY AREA ANALYSIS AND PARAMETER ESTIMATION FOR FOU PROCESSES VIA NON-GEOMETRIC ROUGH PATH THEORY^*

LÉVY AREA ANALYSIS AND PARAMETER ESTIMATION FOR FOU PROCESSES VIA NON-GEOMETRIC ROUGH PATH THEORY^*

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