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.