Splitting of conditional expectations and liftings in product spaces
Positivity๏ผ2024๏ผ
Abstract
Let (X, ๐,P) and (Y, ๐
,Q) be two probability spaces and R be their skew product on the product ฯ -algebra ๐โ๐
. Moreover, let {(๐_y,S_y):yโY} be a Q -disintegration of R (if ๐_y=๐ for every yโY , then we have a regular conditional probability on ๐ with respect to Q ) and let โญ be a sub- ฯ -algebra of ๐โฉโ _yโY๐_y . We prove that if fโโ^โ(R) and ๐ผ_โญโ๐
(f) is the conditional expectation of f with respect to โญโ๐
, then for Q -almost every yโY the y -section [๐ผ_โญโ๐
(f)]^y is a version of the conditional expectation of f^y with respect โญ and S_y . Moreover there exist a lifting ฯ on โ^โ(R) ( R is the completion of R ) and liftings ฯ _y on โ^โ(S_y) , yโ Y , such that [ฯ (f)]^y= ฯ _y ([ฯ (f)]^y ) for all yโ Y and fโโ^โ(R). Both results are generalizations of Strauss et al. (Ann Prob 32:2389โ2408, 2004), where ๐ was assumed to be separable in the Frechet-Nikodรฝm pseudometric and of Macheras et al. (J Math Anal Appl 335:213โ224, 2007), where R was assumed to be absolutely continuous with respect to the product measure PโQ . Finally a characterization of stochastic processes possessing an equivalent measurable version is presented.
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Key words
Disintegration,Regular conditional probability,Conditional expectation,Lifting,Measurable modification of stochastic processes
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