Convergence and parameter estimation of the linear weighted-fractional self-repelling diffusion

Abstract

Let $B^{a,b}$ be a weighted-fractional Brownian motion with Hurst indexes a and b such that $a>-1$ and $0<b<1\wedge (1+a).$ In this paper, we consider the linear self-repelling diffusion $$d{X}_t^{a,b}=d{B}_t^{a,b}+(\theta {\int }_0^t(X_t^{a,b}-X_s^{a,b})\mathrm{ds}+\nu )\mathrm{dt}$$ with $X_0^{a,b}=0,$ where $\theta >0,\nu \in \mathbb{R}$ are two real parameters. The process is an analogue of the linear self-interacting diffusion (Cranston and Le Jan, Math. Ann. 303 (1995), 87-93). We introduce its large time behaviors, and the behavior presents a recursive convergence which is quite different from the asymptotic behavior of stochastic differential equations without interacting drifts. As a related question, we also consider the asymptotic behaviors of the least squares estimations of θ and ν.

KEYWORDS:

• Weighted-fractional Brownian motion
• self-repelling diffusion
• asymptotic normality
• least squares estimation
• Cauchy’s distribution

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 60G22
• 60H07
• 60F05
• 62M09

Additional information

Funding

This project was sponsored by the NSFC (No. 11971101).