Vol. 41, No. 1, 1972

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 307: 1
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Subscriptions
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Author Index
To Appear
 
Other MSP Journals
Conditional expectations associated with stochastic processes

R. A. Brooks

Vol. 41 (1972), No. 1, 33–42
Abstract

A stochastic process, E[x(t,)|ℱt](ω), on a probability space ,.𝒜,P) and an interval D, where x L1(D × Ω) and {ℱt,t D} is an increasing collection of sigma-fields in 𝒜, is considered. Sufficient conditions for the joint measurability of E[x(t,)|ℱt](ω) in (t,ω) are given, and if x L2(D × Ω), it is shown that, under certain fairly general conditions, E[x(t,)|ℱt](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
Authors
R. A. Brooks