THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT

doi:10.1007/s10473-024-0216-x

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (2): 671-685.doi: 10.1007/s10473-024-0216-x

THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT

Xiaoyu XIA¹, Litan YAN^2,*, Qing YANG²

1. College of Information Science and Technology, Donghua University, Shanghai 201620, China;
2. Department of Statistics, College of Science, Donghua University, Shanghai 201620, China

收稿日期:2022-06-21 修回日期:2023-09-27 出版日期:2024-04-25 发布日期:2024-04-16
通讯作者: *Litan YAN, E-mail: litan-yan@hotmail.com
作者简介:Xiaoyu XIA, E-mail: xxiaoyu0617@163.com; Qing YANG, E-mail: qingyang0106@163.com
基金资助:
Yan's work was supported by the NSFC (11971101).

THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT

Xiaoyu XIA¹, Litan YAN^2,*, Qing YANG²

1. College of Information Science and Technology, Donghua University, Shanghai 201620, China;
2. Department of Statistics, College of Science, Donghua University, Shanghai 201620, China

Received:2022-06-21 Revised:2023-09-27 Online:2024-04-25 Published:2024-04-16
Contact: *Litan YAN, E-mail: litan-yan@hotmail.com
About author:Xiaoyu XIA, E-mail: xxiaoyu0617@163.com; Qing YANG, E-mail: qingyang0106@163.com
Supported by:
Yan's work was supported by the NSFC (11971101).

摘要/Abstract

摘要： Let $B^{H} $ be a fractional Brownian motion with Hurst index $\frac{1}{2}\leq H< 1$. In this paper, we consider the equation (called the Ornstein-Uhlenbeck process with a linear self-repelling drift) $\begin{equation*} {\rm d}X_{t}^{H}={\rm d}B_{t}^{H}+\sigma X_t^{H}{\rm d}t+\nu {\rm d}t-\theta \left(\int_{0}^{t}(X_t^{H}-X_{s}^{H}){\rm d}s\right){\rm d}t, \end{equation*} $ where $\theta<0$, $\sigma,\nu \in \mathbb{R}$. The process is an analogue of {self-attracting} diffusion (Cranston, Le Jan. Math Ann, 1995, 303: 87-93). Our main aim is to study the large time behaviors of the process. We show that the solution $X^H$ diverges to infinity as $t$ tends to infinity, and obtain the speed at which the process $X^H$ diverges to infinity as $t$ tends to infinity.

关键词: fractional Brownian motion, stochastic difference equations, rate of convergence, asymptotic

Abstract: Let $B^{H} $ be a fractional Brownian motion with Hurst index $\frac{1}{2}\leq H< 1$. In this paper, we consider the equation (called the Ornstein-Uhlenbeck process with a linear self-repelling drift) $\begin{equation*} {\rm d}X_{t}^{H}={\rm d}B_{t}^{H}+\sigma X_t^{H}{\rm d}t+\nu {\rm d}t-\theta \left(\int_{0}^{t}(X_t^{H}-X_{s}^{H}){\rm d}s\right){\rm d}t, \end{equation*} $ where $\theta<0$, $\sigma,\nu \in \mathbb{R}$. The process is an analogue of {self-attracting} diffusion (Cranston, Le Jan. Math Ann, 1995, 303: 87-93). Our main aim is to study the large time behaviors of the process. We show that the solution $X^H$ diverges to infinity as $t$ tends to infinity, and obtain the speed at which the process $X^H$ diverges to infinity as $t$ tends to infinity.

Key words: fractional Brownian motion, stochastic difference equations, rate of convergence, asymptotic

中图分类号:

60G22

Xiaoyu XIA, Litan YAN, Qing YANG. THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT[J]. 数学物理学报(英文版), 2024, 44(2): 671-685.

Xiaoyu XIA, Litan YAN, Qing YANG. THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT[J]. Acta mathematica scientia,Series B, 2024, 44(2): 671-685.

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THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT

THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT

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