Solvable Filtering Models With O-U Process As Noise

How to solve a filtering equation is a very important part of Stochastic Filtering Theory. It is important for researchers to find which kinds of filtering equation are solvable and how to give representation of the optimal filter. We know the Kalman-Bucy filter is solvable and we only need to get the conditional mean and conditional covariance matrix. However, it is difficult to give the explicit solution of the linear singular filtering model and nonlinear filtering model, which are studying by many researchers.Based on [1], [4] and [7], this paper gives some solvable models and how to solve those models.In§1, we consider two linear models, which are solvable. They are governed by the following stochastic differential equations:where O_t is an m-dimensional O-U process governed by the following SDE:H,b₁,b₀,C are m×d, d×d, d×1, d×d matrices, (B_t, W_t) is a (d+m) dimensional Brownian motion.Another model iswhere Ot is an m-dimensional O-U process governed by the following SDE:H, b₁,b₀, C, A, D are m×d, d×d, d×1, d×k, m×m, m×k dimensional matrices, B_t is a k dimensional Brownian motion.In§2, we consider a nonlinear filtering problem with O-U process as the observation noise. The filter model is given by the following equations:where dO_t=-O_tdt + dW_t,W_t is a 1-dimensional Brownian motion, andμ_i,σ_i are constant.