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STATISTICAL CAUSALITY AND STABLE SUBSPACES OF $H^P$

Part of: Stochastic processes

Published online by Cambridge University Press:  01 August 2012

LJILJANA PETROVIĆ  and
DRAGANA VALJAREVIĆ
LJILJANA PETROVIĆ
Affiliation:
Department of Mathematics and Statistics, Faculty of Economics, University of Belgrade, Kamenička 6, 11000 Beograd, Serbia (email: [email protected])
DRAGANA VALJAREVIĆ*
Affiliation:
Department of Mathematics, Faculty of Science, University of Priština-Kosovska Mitrovica, Lole Ribara 29, 38220 Kosovska Mitrovica, Serbia (email: [email protected])
*
✉For correspondence; e-mail: dragana˙[email protected]
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Abstract

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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.

Keywords

filtrationscausalitystable subspacesextremal measuresstochastic differential equations

MSC classification

Secondary: 60G44: Martingales with continuous parameter
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 1 , August 2013 , pp. 17 - 25
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc. 

References

[1]Bremaud, P. & Yor, M., ‘Changes of filtrations and of probability measures’, Z. Wahrscheinlichkeitstheorie verw. Gebiete 45 (1978), 269–295.CrossRefGoogle Scholar
[2]Gill, J. B. & Petrović, Lj., ‘Causality and stochastic dynamic systems’, SIAM J. Appl. Math. 47(6) (1987), 1361–1366.CrossRefGoogle Scholar
[3]Granger, C. W. J., ‘Investigating causal relations by econometric models and cross spectral methods’, Econometrica 37 (1969), 424–438.CrossRefGoogle Scholar
[4]Grecea, V., ‘On generating $H^p$ spaces of martingales’, Rev. Roumaine Math. Pures Appl. 53(4) (2008), 317–322.Google Scholar
[5]Jacod, J., Calcul Stochastique et Problèmes de Martingales, Lecture Notes in Mathematics (Springer, Berlin, 1979).CrossRefGoogle Scholar
[6]Kunita, H. & Watanabe, S., ‘On square integrable martingales’, Nagoya Math. J. 30 (1967), 209–245.CrossRefGoogle Scholar
[7]Medvegyev, P., Stochastic Integration Theory (Oxford University Press, Oxford, 2007).CrossRefGoogle Scholar
[8]Meyer, P. A., Probability and Potentials (North-Holland, Amsterdam, 1982).Google Scholar
[9]Mykland, P. A., ‘Statistical causality’. Statistical Report No. 14, University of Bergen, 1986.Google Scholar
[10]Mykland, P. A., ‘Stable subspaces over regular solutions of martingale problems’. Statistical Report No 15, University of Bergen, 1986.Google Scholar
[11]Petrović, Lj., ‘Causality and Markovian representations’, Statist. Probab. Lett. 29 (1996), 223–227.CrossRefGoogle Scholar
[12]Petrović, Lj. & Dimitrijević, S., ‘Invariance of statistical causality under convergence’, Statist. Probab. Lett. 81 (2011), 1445–1448.CrossRefGoogle Scholar
[13]Petrović, Lj. & Stanojević, D., ‘Some models of causality and weak solutions of stochastic differential equations with driving semimartingales’, Facta Univ. ser. 20 (2005), 103–112.Google Scholar
[14]Petrović, Lj. & Stanojević, D., ‘Statistical causality, extremal measures and weak solutions of stochastical differential equations with driving semimartingales’, J. Math. Model. Algor. 9 (2010), 113–128.CrossRefGoogle Scholar
[15]Protter, P., Stochastic Integration and Differential Equations (Springer, Berlin, 2004).Google Scholar
[16]Revuz, D. & Yor, M., Continuous Martingales and Brownian Motion (Springer, Berlin, 2005).Google Scholar
[17]Valjarević, D. & Petrović, Lj., ‘Statistical causality and orthogonality of local martingales’, Statist. Probab. Lett. 82 (2012), 1326–1330.CrossRefGoogle Scholar