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Characterisation of Honest Times and Optional Semimartingales of Class- $$(\Sigma )$$ ( Σ )

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  • Libo Li

    (University of New South Wales)

Abstract

Given a finite honest time, we first show that the associated Azéma optional supermartingale can be expressed as the drawdown and the relative drawdown of some local optional supermartingales with continuous running supremum. The relative drawdown representation then allows us to provide a characterisation of finite honest times using a family of non-negative local optional supermartingales with continuous running supremum which converges to zero at infinity. Then we extend the notion of semimartingales of class- $$(\Sigma )$$ ( Σ ) by allowing for jumps in its finite variation part of the semimartingale decomposition. This enables one to establish the Madan–Roynette–Yor option pricing formula for a larger class of processes, and finally, we apply the extended formula to the construction of finite honest times.

Suggested Citation

  • Libo Li, 2022. "Characterisation of Honest Times and Optional Semimartingales of Class- $$(\Sigma )$$ ( Σ )," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2145-2175, December.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:4:d:10.1007_s10959-021-01154-w
    DOI: 10.1007/s10959-021-01154-w
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    References listed on IDEAS

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    1. Kardaras, Constantinos, 2014. "On the characterisation of honest times that avoid all stopping times," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 373-384.
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    6. Madan, D. & Roynette, B. & Yor, Marc, 2008. "Option prices as probabilities," Finance Research Letters, Elsevier, vol. 5(2), pages 79-87, June.
    7. Patrick Cheridito & Ashkan Nikeghbali & Eckhard Platen, 2012. "Processes of Class Sigma, Last Passage Times, and Drawdowns," Published Paper Series 2012-4, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    8. Kardaras, Constantinos, 2015. "On the stochastic behaviour of optional processes up to random times," LSE Research Online Documents on Economics 64965, London School of Economics and Political Science, LSE Library.
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