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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Vector-valued stochastic processes. II. A Radon-Nikodým theorem for vector-valued processes with finite variation


Author: Nicolae Dinculeanu
Journal: Proc. Amer. Math. Soc. 102 (1988), 393-401
MSC: Primary 60G57,; Secondary 60B11,60G20
MathSciNet review: 921006
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Abstract: Given a real-valued process $ A$ with finite variation $ \left\vert A \right\vert$ and a vector-valued process $ B$ with finite variation $ \left\vert B \right\vert$ such that for each $ \omega $, the Stieltjes measure $ dB{\mathbf{.}}(\omega )$ is absolutely continuous with respect to $ dA.(\omega )$, there exists a vector-valued process $ H$ which, under certain separability conditions, satisfies $ {B_t} = \int_{[0,t]} {{H_s}d{A_s}} $ and $ {\left\vert B \right\vert _t} = \int_{[0,t]} {\left\Vert {{H_s}} \right\Vert d{{\left\vert A \right\vert}_s}} $ for every $ t \geq 0$. If, moreover, $ A$ and $ B$ are optional or predictable, then so is $ H$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0921006-4
PII: S 0002-9939(1988)0921006-4
Keywords: Stochastic process, Banach space, Radon-Nikodým, finite variation, increasing process, evanescent, optional, predictable, lifting
Article copyright: © Copyright 1988 American Mathematical Society



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