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A class of remarkable submartingales

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  • Nikeghbali, Ashkan

Abstract

In this paper, we consider the special class of positive local submartingales (Xt) of the form Xt=Nt+At, where the measure is carried by the set {t:Xt=0}. We show that many examples of stochastic processes studied in the literature are in this class and propose a unified approach based on martingale techniques for studying them. In particular, we establish some martingale characterizations for these processes and compute explicitly some distributions involving the pair (Xt,At). We also associate with X a solution to the Skorokhod's stopping problem for probability measures on the positive half-line.

Suggested Citation

  • Nikeghbali, Ashkan, 2006. "A class of remarkable submartingales," Stochastic Processes and their Applications, Elsevier, vol. 116(6), pages 917-938, June.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:6:p:917-938
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    References listed on IDEAS

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    1. Oblój, Jan & Yor, Marc, 2004. "An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale," Stochastic Processes and their Applications, Elsevier, vol. 110(1), pages 83-110, March.
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    Cited by:

    1. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, August.
    2. Fulgence Eyi Obiang & Octave Moutsinga & Youssef Ouknine, 2022. "An Ideal Class to Construct Solutions for Skew Brownian Motion Equations," Journal of Theoretical Probability, Springer, vol. 35(2), pages 894-916, June.
    3. Pospisil, Libor & Vecer, Jan & Hadjiliadis, Olympia, 2009. "Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2563-2578, August.
    4. Ashkan Nikeghbali & Eckhard Platen, 2013. "A reading guide for last passage times with financial applications in view," Finance and Stochastics, Springer, vol. 17(3), pages 615-640, July.
    5. Zhenyu Cui & Duy Nguyen, 2018. "Magnitude and Speed of Consecutive Market Crashes in a Diffusion Model," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 117-135, March.
    6. Fulgence Eyi-Obiang & Youssef Ouknine & Octave Moutsinga, 2017. "On the Study of Processes of $$\sum (H)$$ ∑ ( H ) and $$\sum _\mathrm{s}(H)$$ ∑ s ( H ) Classes," Journal of Theoretical Probability, Springer, vol. 30(1), pages 117-142, March.
    7. Sakrani Samia, 2021. "Time-Changed Local Martingales Under Signed Measures," Journal of Theoretical Probability, Springer, vol. 34(2), pages 644-659, June.
    8. Najnudel, Joseph & Nikeghbali, Ashkan, 2012. "On some universal σ-finite measures related to a remarkable class of submartingales," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1582-1600.
    9. 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.
    10. Oblój, Jan, 2007. "An explicit solution to the Skorokhod embedding problem for functionals of excursions of Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 117(4), pages 409-431, April.
    11. Nikeghbali, Ashkan, 2007. "Non-stopping times and stopping theorems," Stochastic Processes and their Applications, Elsevier, vol. 117(4), pages 457-475, April.
    12. Hongzhong Zhang & Olympia Hadjiliadis, 2012. "Drawdowns and the Speed of Market Crash," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 739-752, September.

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