# Stochastic processes, interpolation of

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; , ). 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 .)