Filtering

Abstract

Let (Ω, F, P) be a probability space and let F _t , t ≥ 0, be a nondecreasing family of sub-σ-algebras of F. Let η(·) be an (Ω, F _t , P) Brownian motion in ℝ^m. Let θ: ω → {1,..., N} be F ₀-measurable with P(θ = j) = $P\left( {\theta = j} \right) = \pi _j^0$; here $\left\{ {\pi _1^0 , \ldots \pi _N^0 } \right\}$ is a fixed but arbitrary distribution on {1,..., N}. For each j = 1,..., N let z _j (·) be a progressively measurable process. Throughout this chapter we shall assume that

$$P\left( {\mathop {\max }\limits_{1 \leqslant j \leqslant N} \int_0^T {\left| {z_j \left( t \right)} \right|^2 dt < \infty ,T > 0} } \right) = 1.$$

((1.1))