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On the Propagation of the Weak Representation Property in Independently Enlarged Filtrations: The General Case

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  • Paolo Tella

    (Technische Universitat Dresden Internationales Hochschulinstitut Zittau)

Abstract

In this paper, we investigate the propagation of the weak representation property (WRP) to an independently enlarged filtration. More precisely, we consider an $${\mathbb {F}}$$ F -semimartingale X possessing the WRP with respect to $${\mathbb {F}}$$ F and an $${\mathbb {H}}$$ H -semimartingale Y possessing the WRP with respect to $${\mathbb {H}}$$ H . Assuming that $${\mathbb {F}}$$ F and $${\mathbb {H}}$$ H are independent, we show that the $${\mathbb {G}}$$ G -semimartingale $$Z=(X,Y)$$ Z = ( X , Y ) has the WRP with respect to $${\mathbb {G}}$$ G , where $${\mathbb {G}}:={\mathbb {F}}\vee {\mathbb {H}}$$ G : = F ∨ H . In our setting, X and Y may have simultaneous jump-times. Furthermore, their jumps may charge the same predictable times. This generalizes all available results about the propagation of the WRP to independently enlarged filtrations.

Suggested Citation

  • Paolo Tella, 2022. "On the Propagation of the Weak Representation Property in Independently Enlarged Filtrations: The General Case," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2194-2216, December.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:4:d:10.1007_s10959-021-01145-x
    DOI: 10.1007/s10959-021-01145-x
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    References listed on IDEAS

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    1. Amendinger, Jürgen, 2000. "Martingale representation theorems for initially enlarged filtrations," Stochastic Processes and their Applications, Elsevier, vol. 89(1), pages 101-116, September.
    2. Di Tella, Paolo, 2020. "On the weak representation property in progressively enlarged filtrations with an application in exponential utility maximization," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 760-784.
    3. El Karoui, Nicole & Jeanblanc, Monique & Jiao, Ying, 2010. "What happens after a default: The conditional density approach," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1011-1032, July.
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