Stochastic processes, interpolation of

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The problem of estimating the values of a stochastic process on some interval using its observed values outside this interval. Usually one has in mind the interpolation estimator for which the mean-square error of interpolation is minimal compared to all other estimators:

the interpolation is called linear if one restricts attention to linear estimators. One of the first problems posed and solved was that of linear interpolation of the value of a stationary sequence. This problem is analogous to the following one: In the space of square-integrable functions on the interval , one must find the projection of onto the subspace generated by the functions , . This problem has been greatly generalized in the theory of stationary stochastic processes (cf. Stationary stochastic process; [1], [2]). One application is the problem of interpolation of the stochastic process arising from the system

where is a linear differential operator of order , and , , is a white noise process. For given initial values , , independent of the white noise, the optimal interpolation estimator , , is the solution of the corresponding boundary value problem

where denotes the formal adjoint operator,

with boundary conditions at the boundary points . For systems of stochastic differential equations the problem of interpolation of some components given the values of other observed components reduces to similar interpolation equations. (See [3].)


[1] A.N. Kolmogorov, "Stationary sequences in Hilbert space" Byull. Moskov. Gos. Univ. Sekt. A , 2 : 6 (1941) pp. 1–40 (In Russian)
[2] Yu.A. Rozanov, "Stationary stochastic processes" , Holden-Day (1967) (Translated from Russian)
[3] R.S. Liptser, A.N. Shiryaev, "Statistics of stochastic processes" , 1–2 , Springer (1977–1978) (Translated from Russian)


The interpolation problem is usually defined as the estimation of an unobserved stochastic process on some time interval given a related stochastic process that is observed outside this time interval. One distinguishes two special cases: 1) linear least-squares interpolation, in which the estimator is constrained to be linear and minimizes a least-squares criterion, see [a1], [a3]; and 2) interpolation in which the conditional distribution of the estimator given the observations is determined, see [a2].

For the Western literature on interpolation see [a5], Sect. 5.3 and [a1], Sect. 4.13. Additional Russian references that have been translated are [a6]; [a7], Sect. 37. For recent developments using stochastic realization theory see [a3], [a4]. Results for the interpolation problem may also be deduced from those for the smoothing problem [a2].


[a1] H. Dym, H.P. McKean, "Gaussian processes, function theory, and the inverse spectral problem" , Acad. Press (1976)
[a2] E. Pardoux, "Equations du filtrage nonlinéaire, de la prédiction et du lissage" Stochastics , 6 (1982) pp. 193–231
[a3] M. Pavon, "New results on the interpolation problem for continuous-time stationary increment processes" SIAM J. Control Optim. , 22 (1984) pp. 133–142
[a4] M. Pavon, "Optimal interpolation for linear stochastic systems" SIAM J. Control Optim. , 22 (1984) pp. 618–629
[a5] N. Wiener, "Extrapolation, interpolation, and smoothing of stationary time series: with engineering applications" , M.I.T. (1949)
[a6] A.M. Yaglom, "Extrapolation, interpolation and filtration of stationary random processes with rational spectral density" Amer. Math. Soc. Sel. Transl. Math. Statist. , 4 (1963) pp. 345–387 Tr. Moskov. Mat. Obshch. , 4 (1955) pp. 333–374
[a7] A.M. Yaglom, "An introduction to the theory of stationary random functions" , Prentice-Hall (1962) (Translated from Russian)
How to Cite This Entry:
Stochastic processes, interpolation of. Encyclopedia of Mathematics. URL:,_interpolation_of&oldid=17488
This article was adapted from an original article by Yu.A. Rozanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article