Maximum Likelihood Estimation in the Fractional Vasicek Model

Abstract

We consider the fractional Vasicek model of the form dXt = (α-βXt)dt +γdBHt , driven by fractional Brownian motion BH with Hurst parameter H ∈ (1/2,1). We construct the maximum likelihood estimators for unknown parameters α and β, and prove their consistency and asymptotic normality.

Maximum Likelihood Estimation for the Fractional Vasicek Model

“…Taking the derivatives of the log-likelihood function with respect to κ and α and setting them to zero, Lohvinenko and Ralchenko (2017) obtained the following expressions for the MLE of α and κ:…”

“…Using the idea of Kleptsyna and Le Breton (2002), Lohvinenko and Ralchenko (2017) obtained the following results:…”

“…In this section, inspired by Lohvinenko and Ralchenko (2017), we extend the asymptotic properties ofα T andκ T from the case of H ∈ (1/2, 1) to the case of H ∈ (0, 1/2]. For the sake of comparison, we first introduce the main result of Lohvinenko and Ralchenko (2017). When H > 1/2, Lohvinenko and Ralchenko (2017) obtained the asymptotic normality for the MLE of α and κ, i.e.,…”

“…Using the Malliavin calculus, Es-Sebaiy and Viens (2019) studied the estimation problem for the drift parameter for some stochastic differential equations driven by fBm. Lohvinenko and Ralchenko (2017) considered the maximum likelihood (ML) estimates of κ and α when κ > 0 and H ∈ (1/2, 1). Moreover, Lohvinenko and Ralchenko (2019) considered the maximum likelihood (ML) estimates of κ and α when κ < 0 and H ∈ (1/2, 1).…”

Maximum likelihood estimation in the non-ergodic fractional Vasicek model

“…Applying the analog of the Girsanov formula for a fractional Brownian motion ([18, Theorem 3], see also [19]) and (6), one can obtain the likelihood ratio dP α,β (T ) dP0,0(T ) for the probability measure P α,β (T ) corresponding to our model and the probability measure P 0,0 (T ) corresponding to the model with zero drift [25]:…”