Vol. 41, No. 1, 1972

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Conditional expectations associated with stochastic processes

R. A. Brooks

Vol. 41 (1972), No. 1, 33–42

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
Received: 28 January 1971
Published: 1 April 1972
R. A. Brooks