The Laws of Large Numbers Associated with the Linear Self-attracting Diffusion Driven by Fractional Brownian Motion and Applications

Let $$B^H$$ B H be a fractional Brownian motion with Hurst index $$\frac{1}{2}\le H 0$$ θ > 0 and $$\nu \in {{\mathbb {R}}}$$ ν ∈ R are two unknown parameters. The process is an analogue of the linear self-attracting diffusion (Cranston and Le Jan in Math Ann 303:87–93, 1995). We introduce some laws of large numbers associated with the self-attracting diffusion. In particular, we show that the convergences $$\begin{aligned} \frac{1}{\theta T^{3-2H}}\int _0^TY^H_t\mathrm{d}B^H_t,\quad \frac{1}{T^{3-2H}}\int _0^T\left( Y^H_t\right) ^2\mathrm{d}t\longrightarrow \frac{H\theta ^{-2H}}{3-2H} \Gamma (2H) \end{aligned}$$ 1 θ T 3 - 2 H ∫ 0 T Y t H d B t H , 1 T 3 - 2 H ∫ 0 T Y t H 2 d t ⟶ H θ - 2 H 3 - 2 H Γ ( 2 H ) hold almost surely and in $$L^2(\Omega )$$ L 2 ( Ω ) , as $$T\rightarrow \infty $$ T → ∞ , where $$Y^H_t=\int ^t_0(X^H_t-X^H_s)\mathrm{d}s$$ Y t H = ∫ 0 t ( X t H - X s H ) d s and the integral $$\int _0^TY^H_t\mathrm{d}B^H_t$$ ∫ 0 T Y t H d B t H is the Young integral. As some applications, we study asymptotic behaviors of the least squares estimators of $$\theta $$ θ and $$\nu $$ ν under the continuous observation.