Abstract
We investigate the fractional Vasicek model described by the stochastic differential equation $dX_t=(\alpha -\beta X_t)\,dt+\gamma \,dB^H_t$, $X_0=x_0$, driven by the fractional Brownian motion $B^H$ with the known Hurst parameter $H\in (1/2,1)$. We study the maximum likelihood estimators for unknown parameters $\alpha$ and $\beta$ in the non-ergodic case (when $\beta <0$) for arbitrary $x_0\in \mathbb{R}$, generalizing the result of Tanaka, Xiao and Yu (2019) for particular $x_0=\alpha /\beta$, derive their asymptotic distributions and prove their asymptotic independence.
Highlights
The present paper deals with the fractional Vasicek model of the form dXt = (α − βXt )dt + γ dBtH, X0 = x0 ∈ R, (1)where BH is the fractional Brownian motion with the Hurst index H ∈ (1/2, 1)
Where BH is the fractional Brownian motion with the Hurst index H ∈ (1/2, 1). It is a generalization of the classical interest rate model proposed by O
In order to use this model in practice, a theory of parameter estimation is necessary
Summary
The least squares and ergodic-type estimators of unknown parameters α and β were studied in [27, 38, 39]. The corresponding MLEs of α and β were presented in [25] Their consistency and asymptotic normality were proved there for the case β > 0. More general results were proved in [26], where joint asymptotic normality of MLE of the vector parameter (α, β) was established. Tanaka et al [33] investigated asymptotic behavior of MLEs in the cases β = 0 and β < 0. The asymptotic behavior of the process X and of the estimators substantially depends on the sign of the parameter β. The asymptotic behavior of the MLEs in this case requires a separate study. Some auxiliary facts and results concerning modified Bessel functions of the first kind and MGFs related to the normal distribution are collected in the appendices
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