Conditional expectations associated with stochastic processes

R. A. Brooks

Vol. 41 (1972), No. 1, 33–42

DOI: 10.2140/pjm.1972.41.33

Abstract

A stochastic process, E[x(t,⋅)|ℱ_t](ω), on a probability space (Ω,.𝒜,P) and an interval D, where x ∈ L₁(D × Ω) and {ℱ_t,t ∈ D} is an increasing collection of sigma-fields in 𝒜, is considered. Sufficient conditions for the joint measurability of E[x(t,⋅)|ℱ_t](ω) in (t,ω) are given, and if x ∈ L₂(D × Ω), it is shown that, under certain fairly general conditions, E[x(t,⋅)|ℱ_t](ω) can be identified with the projection of x onto a certain subspace of the Hilbert space associated with L₂(D × Ω). The results obtained herein have application in certain classes of stochastic optimization problems.

Mathematical Subject Classification 2000

Primary: 60G05

Milestones

Received: 28 January 1971

Published: 1 April 1972