We deal with the problem of optimal estimation of the linear functionals constructed from the missed values of a continuous time stochastic process $\xi(t)$ with periodically stationary increments at points $t\in[0;(N+1)T]$ based on observations of this process with periodically stationary noise. To solve the problem, a sequence of stochastic functions $ \{\xi^{(d)}_j(u)=\xi^{(d)}_j(u+jT,\tau),\,\, u\in [0,T), j\in\mathbb Z\}. $ is constructed. It forms a $L_2([0,T);H)$-valued stationary increment sequence $\{\xi^{(d)}_j,j\in\mathbb Z\}$ or corresponding to it an (infinite dimensional) vector stationary increment sequence $\{\vec\xi^{(d)}_j=(\xi^{(d)}_{kj}, k=1,2,\dots)^{\top}, j\in\mathbb Z\}$. In the case of a known spectral density, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.

翻译：本文研究了基于带有周期平稳噪声的连续时间随机过程$\xi(t)$在$t\in[0;(N+1)T]$范围内的观测，对该过程在点$t$处具有周期平稳增量的缺失值所构成的线性泛函进行最优估计的问题。为求解该问题，构造了一组随机函数序列$\{\xi^{(d)}_j(u)=\xi^{(d)}_j(u+jT,\tau),\,\, u\in [0,T), j\in\mathbb Z\}$，该序列形成$L_2([0,T);H)$值平稳增量序列$\{\xi^{(d)}_j,j\in\mathbb Z\}$或对应的（无穷维）向量平稳增量序列$\{\vec\xi^{(d)}_j=(\xi^{(d)}_{kj}, k=1,2,\dots)^{\top}, j\in\mathbb Z\}$。在已知谱密度的情况下，推导出计算泛函最优估计的均方误差值及谱特征的公式。在给定可容许谱密度集合的情况下，导出了确定最小不利谱密度及泛函最优线性估计的最小最大（鲁棒）谱特征的公式。