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STATISTICAL CAUSALITY AND STABLE SUBSPACES OF
$H^P$
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 88 / Issue 1 / August 2013
- Published online by Cambridge University Press:
- 01 August 2012, pp. 17-25
- Print publication:
- August 2013
-
- Article
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In this paper we consider the statistical concept of causality in continuous time between filtered probability spaces, based on Granger’s definitions of causality. Then we consider some stable subspaces of
$H^p$ which contain right continuous modifications of martingales 
$P(A \mid {\mathcal {G}}_t)$. We give necessary and sufficient conditions, in terms of statistical causality, for these spaces to coincide with 
$H^p$. These results can be applied to extremal measures and regular weak solutions of stochastic differential equations.
