Abstract
In the paper, we tackle the least squares estimators of the Vasicek-type model driven by sub-fractional Brownian motion: d X t = ( μ + θ X t ) d t + d S t H , t ≥ 0 with X 0 = 0 , where S H is a sub-fractional Brownian motion whose Hurst index H is greater than 1 2 , and μ ∈ R , θ ∈ R + are two unknown parameters. Based on the so-called continuous observations, we suggest the least square estimators of μ and θ and discuss the consistency and asymptotic distributions of the two estimators.
Highlights
Statistical inference for stochastic equations is a main research direction in probability theory and its applications
We mention the works of Berzin et al [6], Es-Sebaiy [7], Es-Sebaiy and Nourdin [8], Hu and Nualart et al [9,10], Kleptsyna and Le Breton [11], Prakasa Rao [12], and the references therein for results on parameter estimation of stochastic equations driven by the fractional Brownian motion
We discuss the least squares estimation for the Vasicek-type model driven by a sub-fraction Brownian motion S H with Hurst index H ∈ ( 12, 1)
Summary
Statistical inference for stochastic equations is a main research direction in probability theory and its applications. For μ = 0 and G a Gaussian process, El Machkouri et al [14] showed the strong consistency and the asymptotic distribution of the least squares estimator θof θ based on the properties of G, and as some examples, the authors studied the three Vasicek-type models driven by fractional Brownian motion, sub-fractional Brownian motion, and bi-fractional Brownian motion, respectively Motivated by these above results and for simplicity, in this paper, we consider the least squares estimation of Equation (1) when G is a sub-fractional Brownian motion S H with Hurst index H ∈ ( 12 , 1).
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