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].)

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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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