Bulletin of the Australian Mathematical Society

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Research Article

STATISTICAL CAUSALITY AND STABLE SUBSPACES OF H P

LJILJANA PETROVIĆa1 and DRAGANA VALJAREVIĆa2 c1

a1 Department of Mathematics and Statistics, Faculty of Economics, University of Belgrade, Kamenička 6, 11000 Beograd, Serbia (email: petrovl@ekof.bg.ac.rs)

a2 Department of Mathematics, Faculty of Science, University of Priština-Kosovska Mitrovica, Lole Ribara 29, 38220 Kosovska Mitrovica, Serbia (email: dragana_stan@yahoo.com)

Abstract

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∣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.

(Received May 17 2012)

(Accepted May 23 2012)

2010 Mathematics subject classification

  • primary 60G44; secondary 60H10

Keywords and phrases

  • filtrations;
  • causality;
  • stable subspaces;
  • extremal measures;
  • stochastic differential equations

Correspondence:

c1 For correspondence; e-mail: dragana˙stan@yahoo.com

Footnotes

The work is supported by the Serbian Ministry of Science and Technology (Grants 044006 and 179005).