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On the optional and orthogonal decompositions of supermartingales and applications

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  • Berkaoui, Abdelkarem

Abstract

We consider a set Q of probability measures, which are absolutely continuous with respect to the physical probability measure P and at least one is equivalent to P. We investigate in the finite probability space case, necessary and sufficient conditions on Q, under which any Q-supermartingale can be decomposed into the sum of a Q-martingale and a decreasing process. We also provide an orthogonal decomposition of Q-super-martingales and apply this result to the orthogonal decomposition of the polar cone of Q.

Suggested Citation

  • Berkaoui, Abdelkarem, 2023. "On the optional and orthogonal decompositions of supermartingales and applications," Statistics & Probability Letters, Elsevier, vol. 199(C).
    • Handle: RePEc:eee:stapro:v:199:y:2023:i:c:s0167715223000743
    DOI: 10.1016/j.spl.2023.109850
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    References listed on IDEAS

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    1. S. D. Jacka, 1992. "A Martingale Representation Result and an Application to Incomplete Financial Markets," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 239-250, October.
    2. Föllmer, Hans & Kramkov, D. O., 1997. "Optional decompositions under constraints," SFB 373 Discussion Papers 1997,31, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    3. Ioannis Karatzas & Constantinos Kardaras, 2015. "Optional Decomposition for continuous semimartingales under arbitrary filtrations," Papers 1501.04274, arXiv.org, revised Feb 2015.
    4. Choulli, Tahir & Vandaele, Nele & Vanmaele, Michèle, 2010. "The Föllmer-Schweizer decomposition: Comparison and description," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 853-872, June.
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