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 ∧ ( 1 + a ) . In this paper, we consider the linear self-repelling diffusion d X t a , b = d B t a , b + ( θ ∫ 0 t ( X t a , b − X s a , b ) ds + ν ) dt with X 0 a , b = 0 , where θ > 0 , ν ∈ 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.