# Stochastic processes, interpolation of

The problem of estimating the values of a stochastic process $X ( t)$ on some interval $a < t < b$ using its observed values outside this interval. Usually one has in mind the interpolation estimator $\widehat{X} ( t)$ for which the mean-square error of interpolation is minimal compared to all other estimators:

$${\mathsf E} | \widehat{X} ( t) - X ( t) | ^ {2} = \min ;$$

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 $X ( 0)$ of a stationary sequence. This problem is analogous to the following one: In the space $L _ {2}$ of square-integrable functions on the interval $- \pi < \lambda \leq \pi$, one must find the projection of $\phi ( \lambda ) \in L _ {2}$ onto the subspace generated by the functions $e ^ {i \lambda k } \phi ( \lambda )$, $k = \pm 1 , \pm 2 ,\dots$. 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

$$L X ( t) = Y ( t) ,\ t > t _ {0} ,$$

where $L$ is a linear differential operator of order $l$, and $Y ( t)$, $t > t _ {0}$, is a white noise process. For given initial values $X ^ {(} k) ( t _ {0} )$, $k = 1 \dots l - 1$, independent of the white noise, the optimal interpolation estimator $X ( t)$, $a < t < b$, is the solution of the corresponding boundary value problem

$$L ^ {*} L \widehat{X} ( t) = 0 ,\ \ a < t < b ,$$

where $L ^ {*}$ denotes the formal adjoint operator,

$$\widehat{X} {} ^ {(} k) ( s) = X ^ {(} k) ( s) ,\ \ k = 0 \dots l ,$$

with boundary conditions at the boundary points $s = a , b$. 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)