Space-differential filtration of Markov processes on one-dimensional stochastic manifolds (Q1913109)

Let the dynamics of an object be described by \[ dL = F_L(L,t) dt + G_L (t) dV_L (t), \] where \(F_L \in \mathbb{R}^n\), \(G_L \in \mathbb{R}^{n \times p}\), \(V_L \in \mathbb{R}^p\) is the standard Wiener process, \(L^T = (l_1, \dots, l_n)\), and \(l_1\) is the natural parameter with the assumption \(\text{Pr} (\dot l_1 \leq 0) \ll 1\). Suppose that the observations \(Y_i\) at discrete times \(t_i\) are determined by \[ Y_i = S[Z (l_{1i}), t_i] + N_i \] \((i = 1,2,\dots)\) where \(S\) is a known linear continuous vector function, the \(N_i\) are Gaussian, and \(Z \in \mathbb{R}^m\) is prescribed by \[ dZ = F_Z(Z,l_1) dl_1 + G_Z(Z, l_1) dV_Z (l_1), \] where \(F_Z \in \mathbb{R}^m\), \(G_Z \in \mathbb{R}^{m \times q}\), and \(V_Z \in \mathbb{R}^q\) is the standard Wiener process. \(V_Z\), \(V_L\) and \(N_i\) are independent of each other. The author formulates the principles of space-differential filtering for the above problem. A series of equations are set up for extrapolation and inverse interpolation of the process and for forming an optimal filtering algorithm. Particularly, under certain conditions, and using the first-order Gaussian approximation, the synthesis of a filtering algorithm is rather simple.