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On the stochastic behaviour of optional processes up to random times

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  • Kardaras, Constantinos

Abstract

In this paper, a study of random times on filtered probability spaces is undertaken. The main message is that, as long as distributional properties of optional processes up to the random time are involved, there is no loss of generality in assuming that the random time is actually a randomised stopping time. This perspective has advantages in both the theoretical and practical study of optional processes up to random times. Applications are given to financial mathematics, as well as to the study of the stochastic behaviour of Brownian motion with drift up to its time of overall maximum as well as up to last-passage times over finite intervals. Furthermore, a novel proof of the Jeulin–Yor decomposition formula via Girsanov’s theorem is provided.

Suggested Citation

  • 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.
    • Handle: RePEc:ehl:lserod:64965
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    File URL: http://eprints.lse.ac.uk/64965/
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    References listed on IDEAS

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    1. R. J. Elliott & M. Jeanblanc & M. Yor, 2000. "On Models of Default Risk," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 179-195, April.
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    Cited by:

    1. Libo Li & Ruyi Liu & Marek Rutkowski, 2022. "Vulnerable European and American Options in a Market Model with Optional Hazard Process," Papers 2212.12860, arXiv.org.
    2. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, August.
    3. Acciaio, Beatrice & Fontana, Claudio & Kardaras, Constantinos, 2016. "Arbitrage of the first kind and filtration enlargements in semimartingale financial models," LSE Research Online Documents on Economics 65150, London School of Economics and Political Science, LSE Library.
    4. 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.
    5. Acciaio, Beatrice & Fontana, Claudio & Kardaras, Constantinos, 2016. "Arbitrage of the first kind and filtration enlargements in semimartingale financial models," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1761-1784.
    6. Černý, Aleš & Ruf, Johannes, 2023. "Simplified calculus for semimartingales: Multiplicative compensators and changes of measure," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 572-602.

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    More about this item

    Keywords

    Random times; randomised stopping times; times of maximum last; passage times;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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