Adapted probability distributions. (English) Zbl 0548.60019

Let x be a stochastic process on a probability space endowed with a filtration \(FB_ t\), \(t\geq 0\). Consider the family of new stochastic processes in finitely many time variables obtained from x by iterating the following two operations: (1) composition by continuous functions, and (2) conditional expectation with respect to \(FB_ t\). Two stochastic processes x and y are said to have the same adapted distribution if each pair of new processes obtained from x and y by iterating (1) and (2) have the same finite dimensional distributions. By restricting to n iterations of (2) one obtains a sequence of stronger and stronger equivalence relations \(x\equiv_ ny\), \(n=0,1,2,..\). between stochastic processes. For each \(n\equiv_{n+1}\) strictly refines \(\equiv_ n\). The weakest relation \(x\equiv_ 0y\) amounts to identity of the finite dimensional distributions of x and y. The next relation \(x\equiv_ 1y\) amounts to synonimity of x and y (in the sense of D. Aldous). Finally, the strongest relation, \(x\equiv_ ny\) for all n, means that x and y have the same adapted distribution.
Analysis of the strongest equivalence relation leads to probability spaces with a strong universality property for adapted stochastic processes, called saturation. On probability spaces having this property, there exist ’strong’ solutions to a large class of stochastic integral equations.
Reviewer: M.Iosifescu


60E05 Probability distributions: general theory
60G05 Foundations of stochastic processes
60A10 Probabilistic measure theory
60G48 Generalizations of martingales
03B48 Probability and inductive logic
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