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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On a subclass of square integrable martingales


Author: Dean Isaacson
Journal: Proc. Amer. Math. Soc. 34 (1972), 521-526
MSC: Primary 60J65
DOI: https://doi.org/10.1090/S0002-9939-1972-0295432-1
MathSciNet review: 0295432
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Abstract: Let $ \mathcal{M}_2^ \ast $ denote the class of continuous, nowhere constant, square integrable martingales, $ M(t) = X({\langle M\rangle _t})$, for which $ {\langle M\rangle _t}$ is a time change on the $ \sigma $-fields generated by the Brownian motion $ X(t)$. It is shown that if $ M(t) \in \mathcal{M}_2^ \ast $, then the family of $ \sigma $-fields generated by $ M(t)$ is a right continuous family. If $ M(t) \in \mathcal{M}_2^ \ast $ and if $ \sigma \{ M(s):s \leqq t\} = \sigma \{ X(s):s \leqq t\} $ for some Brownian motion $ X(t)$, then $ M(t) = \smallint_0^t {\Phi (s)dX(s)} $ and $ X(t) = \smallint_0^t {(1/\Phi (s))dM(s)} $ for some process $ \Phi (s)$ with $ \Phi (s) \ne 0$ a.e. $ dt \times dP$.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0295432-1
Keywords: Continuous square integrable nowhere constant martingales, Brownian motion, time change, stochastic integral
Article copyright: © Copyright 1972 American Mathematical Society