Minimax interpolation of stochastic processes with stationary increments from observations with noise

The problem of optimal estimation of the linear functional \[ A_T{\xi }=\int _0^Ta(t)\xi (t) dt \] depending on unknown values of the stochastic process $\xi (t)$ with stationary increments from observations of the process $\xi (t)+\eta (t)$ at points $t\in \mathbb R\setminus [0;T]$ is considered, where $\eta (t)$ is a stationary stochastic process uncorrelated with $\xi (t)$. Formulas for calculating the mean square error and spectral characteristic of the optimal linear estimate of the functional are proposed in the case where spectral densities are known. Otherwise relations that determine the least favorable spectral densities and minimax spectral characteristics are proposed for given sets of admissible spectral densities.