### Journals Information

**
Mathematics and Statistics Vol. 2(2), pp. 78 - 88 DOI: 10.13189/ms.2014.020204 Reprint (PDF) (149Kb) **

## Robust Extrapolation Problem for Stochastic Processes with Stationary Increments

**Maksym Luz , Mikhail Moklyachuk ^{*}**

Department of Probability Theory, Statistics and Actuarial Mathematics,Taras Shevchenko National University of Kyiv, Kyiv 01601, Ukraine

**ABSTRACT**

The problem of optimal estimation of the linear functionals and depending on the unknown values of stochastic process ξ(t), t ∈ R, with stationary nth increments from observations of the process at points t < 0 is considered. Formulas for calculating the mean square error and the spectral characteristic of optimal linear estimates of the functionals are derived in the case where the spectral density of the process is exactly known. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristic of the optimal linear estimates of the functionals are proposed in the case where the spectral density of the process is not exactly known, but a set of admissible spectral densities is given.

**KEYWORDS**

Stochastic Process with Stationary Increments, Minimax-Robust Estimate, Mean Square Error, Least Favorable Spectral Density, Minimax-Robust Spectral Characteristic

**Cite this paper**

Maksym Luz , Mikhail Moklyachuk (2014). Robust Extrapolation Problem for Stochastic Processes with Stationary Increments. *Mathematics and Statistics, 2 , * 78 - 88. doi: 10.13189/ms.2014.020204.