Using bases of finite functions in problems of filtering of a priori uncertain time-dependent processes on stochastic spatial fractals (Q1276788)

The author considers a comparatively new class of problems connected with the estimation of time-dependent stochastic processes on one-dimensional spatial stochastic manifolds. The model of a spatial manifold is described by stochastic spatial-differential equation \[ dx= a(x,\widehat{\ell}) d\widehat{\ell}+ q(x,\widehat{\ell}) dw(\widehat{\ell}), \] where \(\widehat{\ell}\) is the natural parameter, \(w\) is a vector-valued spatial Wiener process. The equation of observation has the form \[ y_s= S(x(\widehat{\ell}_s),t_s)+ v_{y^s}, \quad s=1,2,\dots,\quad t_s\in [t_0,t_0+T], \] \(v_{y^s}\) is a vector Gaussian sequence. Some basis consisting of compactly supported functions is choosen on the segment \([t_0,t_0+T]\) and the process \(\widehat{\ell}_t\) is projected on the linear span of this basis. So, the problem of spatial-differential filtering of the a priori indefinite time-dependent process \(\widehat{\ell}\) considered on a one-dimensional stochastic spatial fractal is solved using the basis of compactly supported functions. The maximum likelihood test and the test connected with a posteriori probability density are used for an estimation of the optimality of the filtering procedure. A collection of normalized \(B\)-splines of different powers is considered as an example of the basis of compactly supported functions.