Filtering of stochastic evolution equations. (English. Russian original) Zbl 0639.60050

Makhno, S. Ya.

Theory Probab. Math. Stat. 33, 75-86 (1986); translation from Teor. Veroyatn. Mat. Stat. 33, 68-79 (1985).

The filtering problem is solved for a nonobservable process that is a solution of the equation \[ u(t)=u(0)+\int^{t}_{0}[A(s)u(s)+A_ 1(s,\xi)]ds+\int^{t}_{0}B(s,\xi)dw(s) \] from observations of the process \[ \xi (t)=\xi_ 0+\int^{t}_{0}[a(s,\xi)+f(s)u(s)]ds+\int^{t}_{0}b(s,\xi)dw_ 1(s). \] In these equations A(t) is a monotone operator from a Banach space V into \(V^*\), w(s) is a Wiener process in a Hilbert space E, \(\xi (t)\in R_ n\), and \(w_ 1(s)\) is a Wiener process independent of w(s).

MSC:

60G35	Signal detection and filtering (aspects of stochastic processes)
93E11	Filtering in stochastic control theory
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
35G10	Initial value problems for linear higher-order PDEs
35K25	Higher-order parabolic equations
62M20	Inference from stochastic processes and prediction

Keywords:

bounded semigroup; stochastic partial differential equations; Wiener process in a Hilbert space