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 $\hat{X}(t)$ for which the mean-square error of interpolation is minimal compared to all other estimators:

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

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 $-\pi <\lambda \le \pi$, one must find the projection of $\varphi (\lambda )\in L_2$ onto the subspace generated by the functions $e^{i\lambda k}\varphi (\lambda )$, $k=\pm 1,\pm 2,\dots$. 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

$$LX(t)=Y(t),t>t_0,$$

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

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

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

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

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].)

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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].

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