# 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].)

#### References

 [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)