Conditionally optimal interpolation of random processes defined by stochastic differential equations (Q1595693)

A partially observable stochastic process \(\{Z_t,Y_t\}\) is considered, which is defined by nonlinear stochastic differential equations. The optimal interpolation problem is to obtain an optimal estimate \(\widehat Z_{s/t}\), \(s<t\), of the value \(Z_s\) of the nonobservable process from the results of observations \(\{Y_u,0\leq u\leq t\}\). Efficient solutions of the problem have been obtained only for linear equations and for nonlinear equations of conditional Gaussian processes. \textit{I. N. Sinitsyn} and \textit{V. I. Shin} [Russ. Acad. Sci., Dokl., Math. 49, No. 3, 539-544 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 336, No. 4, 453-456 (1994; Zbl 0828.60027)] developed a theory of interpolation of random sequences defined by nonlinear stochastic difference equations. In the present paper the theory is extended to stochastic processes. Three types of interpolation problems are solved: the fixed point, fixed lag, and fixed interval interpolation. The calculation for constructing a conditionally optimal interpolator can be done before hand, during the designing of an interpolator, with the help of corresponding software, see \textit{Pugachev} et al. [Autom. Remote Control 52, No. 1, 73-80 (1991)]. The interpolation itself can be done on a real-time scale.