Maximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments

Abstract

The paper deals with the regression model X t = θt+B t , t ∈ [0, T ], where B = {B t , t ≥ 0} is a centered Gaussian process with stationary increments. We study the estimation of the unknown parameter θ and establish the formula for the likelihood function in terms of a solution to an integral equation. Then we find the maximum likelihood estimator and prove its strong consistency. The results obtained generalize the known results for fractional and mixed fractional Brownian motion.

Parameter Estimation for Gaussian Processes with Application to the Model with Two Independent Fractional Brownian Motions

“…Theorem 4 (Relations between discrete and continuous MLEs [33]). Let the assumptions of Theorem 2 hold.…”

“…Theorem 2 (Likelihood function and continuous-time MLE [33]). Let T be fixed, assumptions (A)-(C) hold.…”

“…We consider the MLEs for discrete and continuous schemes of observations. The results presented are based on the recent papers [33,32]. Note that in [32] the model (1) with G(t) = t was considered, and the driving process B was a process with stationary increments.…”

“…The results presented are based on the recent papers [33,32]. Note that in [32] the model (1) with G(t) = t was considered, and the driving process B was a process with stationary increments. Then in [33] these results were extended to the case of non-linear drift and more general class of driving processes.…”

“…Note that in [32] the model (1) with G(t) = t was considered, and the driving process B was a process with stationary increments. Then in [33] these results were extended to the case of non-linear drift and more general class of driving processes. In the present paper we apply the theoretical results mentioned above to the models with fractional Brownian motion, mixed fractional Brownian motion and sub-fractional Brownian motion.…”