Properties of maximum likelihood estimates in diffusion and fractional Brownian models. (Ukrainian, English) Zbl 1051.62067

Rudomino-Dusyatska, N.

Teor. Jmovirn. Mat. Stat. 68, 125-132 (2003); translation in Theory Probab. Math. Stat. 68, 139-146 (2004).

The author considers the stochastic differential equation \[ dX_{t}=\theta X_{t}+\sigma_1 X_{t}dW_{t}+\sigma_2 X_{t}dB_{t}^{H}, \] where \(W_{t}\) is a Wiener process and \(B_{t}^{H}\) is a fractional Brownian motion with index \(H\in (1/2,1)\), and \(B_{t}^{H}\) is independent on \(W_{t}\). Two maximum likelihood estimates of the parameter \(\theta\) are constructed and compared. Asymptotic normality and asymptotic effeciency of estimates of the drift parameter \(\theta\) in the models \[ \begin{aligned} dX_{t}&= T^{-\alpha} \theta X_{t}dt+cX_{t}dW_{t},\quad \theta\in R,\;\alpha\in (1/2,1]\\ \text{and} dX_{t} &=T^{-\alpha} \theta X_{t}dt+cX_{t}dB_{t}^{H},\quad \theta\in R,\;\alpha\in (1-H,1] \end{aligned} \] are proved.

Reviewer: A. D. Borisenko (Kyïv)

MSC:

62M05	Markov processes: estimation; hidden Markov models
62F12	Asymptotic properties of parametric estimators
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

maximum likelihood estimate; diffusion model; fractional Brownian model; asymptotic normality; asymptotic effectiveness