Generalized inverse extrapolation of stochastic processes by an aggregate of continuous discrete observations with memory (Q1386990)

In the case of continuous observations the considered system is of the form (in the sense of Ito) \[ dx_t= f(t,x_t)dt +\Phi_1 (t,x_t)dw_t, \] \[ dz_t =h(t,x_t,x_{\tau_1}, \dots, x_{\tau_N }, z_t)dt +\Phi_2 (t,z_t) dv_t, \] where \(x_t\) is an \(n\)-dimensional unobserved process, \(z_t\) is an \(l\)-dimensional observed process, \(0<\tau_N <\tau_{N-1} <\cdots <\tau_1 \leq t\). The authors derive the equations of nonlinear extrapolation for mean square optimal estimations \(\mu(s_k,t)\) of the process values \(x_{s_k}\), \(k=1, \dots,K\), where the fixed time moments \(s_1, \dots, s_K\) are such that \(t<s_1 <s_2< \cdots <s_K\). The case of continuous together with discrete observations is considered as well.