Abstract
In this study, as a continuation to the studies of the self-interaction diffusion driven by subfractional Brownian motion SH, we analyze the asymptotic behavior of the linear self-attracting diffusion:dXtH=dStH−θ∫0t(XtH−XsH)dsdt+νdt,X0H=0,where θ > 0 and ν∈R are two parameters. When θ < 0, the solution of this equation is called self-repelling. Our main aim is to show the solution XH converges to a normal random variable X∞H with mean zero as t tends to infinity and obtain the speed at which the process XH converges to X∞H as t tends to infinity.
Highlights
In a previous study (I), as an extension to classical result, we considered the linear selfinteracting diffusion as follows: ts XHt SHt − θ (XHs − XHu )duds + ]t, t ≥ 0, (1)with θ ≠ 0, where θ and ] are two real numbers, and SH is a sub-fBm with the Hurst parameter ≤ H < The solution ofWhen θ < 0, in a previousEq 1 study is called self-repelling if θ < 0 and (I), we showed that the solution XH is called diverges self-attracting to infinity as t if θ > tends0. to infinity and JH0 (t; θ, ])dte12 θt2 XHt
By Lemma (3.5), 3.3 and the equation almost surely as t tends to infinity, we find that
It follows from the integration by parts that hθ(s)dSHs −hθ(t)SHt − SHs dhθ(s) → 0 t t almost surely as t tends to infinity
Summary
In a previous study (I) (see [12]), as an extension to classical result, we considered the linear selfinteracting diffusion as follows: ts XHt SHt − θ (XHs − XHu )duds + ]t, t ≥ 0,. When H 12, as a special case of path-dependent stochastic differential equations, in 1995, Cranston and Le Jan [8] introduced a linear self-attracting diffusion (Eq 1) with θ > 0. They showed that the process Xt converges in L2 and almost surely as t tends infinity. More examples can be found in Benaïm et al [2, 3], Cranston and Mountford [9], Gan and Yan [11], Gauthier [13], Herrmann and Roynette [14], Herrmann and Scheutzow [15], Mountford and Tarr [20], Sun and Yan [26, 27], Yan et al [34], and the references therein In this present study, our main aim is to expound and prove the following statements:.
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