Optimal Linear Nonstationary Filtering

Abstract

On the probability space (Ω, F, P) with a distinguished family of the σ-algebras (F _t ), t ≤ T, we shall consider the two-dimensional Gaussian random process (θ _t , F _t ), 0 ≤ t ≤ T, satisfying the stochastic differential equations

$$d{\theta _t}\, = \,a(t){\theta _t}dt\, + \,b(t)d{W_1}(t)$$

(10.1)

$$d{\xi _t}\, = \,A(t){\theta _t}dt\, + \,B(t)d{W_2}(t),$$

(10.2)

where W ₁ = (W ₁(t), F _t ) and W ₂ = (W ₂(t), F _t ) are two independent Wiener processes and θ ₀, ξ ₀ are F ₀-measurable.