Maximum likelihood drift estimation for Gaussian process with stationary increments

The paper deals with the regression model \(X_t = \theta t + B_t\), \(t\in[0, T ]\), where \(B=\{B_t, t\geq 0\}\) is a centered Gaussian process with stationary increments. We study the estimation of the unknown parameter \(\theta\) 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.