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SDEs with Uniform Distributions : Peacocks, Conic Martingales and Mean Reverting Uniform Diffusions

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  • BRIGO, Damiano

    (Imperial College London)

  • JEANBLANC, Monique

    (Université d’Evry-Val-d’Essonne)

  • VRINS, Frédéric

    (UCL, Louvain Finance Center)

Abstract

We introduce a way to design Stochastic Differential Equations of diffusion type admitting a unique strong solution distributed as a uniform law with general conic time-boundaries. We show that these processes are new diffusion martingales, hence peacocks, and recover two previously known special cases with square-root and linear time-boundaries. We study local time and activity of such processes. We further introduce general mean-reverting diffusion processes having a uniform law at all times evolving between constant boundaries. This may be used to model random probabilities, random recovery rates or random correlations. We verify via an Euler scheme simulation that they have the desired uniform behavior.

Suggested Citation

  • BRIGO, Damiano & JEANBLANC, Monique & VRINS, Frédéric, 2016. "SDEs with Uniform Distributions : Peacocks, Conic Martingales and Mean Reverting Uniform Diffusions," LIDAM Discussion Papers CORE 2016046, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2016046
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    References listed on IDEAS

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    1. Monique Jeanblanc & Frédéric Vrins, 2018. "Conic martingales from stochastic integrals," Mathematical Finance, Wiley Blackwell, vol. 28(2), pages 516-535, April.
    2. Brigo, Damiano & Jeanblanc, Monique & Vrins, Frédéric, 2020. "SDEs with uniform distributions: Peacocks, conic martingales and mean reverting uniform diffusions," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 3895-3919.
    3. Peter Carr, 2017. "Bounded Brownian Motion," Risks, MDPI, vol. 5(4), pages 1-11, November.
    4. Brigo, Damiano, 2000. "On SDEs with marginal laws evolving in finite-dimensional exponential families," Statistics & Probability Letters, Elsevier, vol. 49(2), pages 127-134, August.
    5. Damiano Brigo & Fabio Mercurio, 2000. "Option pricing impact of alternative continuous-time dynamics for discretely-observed stock prices," Finance and Stochastics, Springer, vol. 4(2), pages 147-159.
    6. Christophe Profeta & Frédéric Vrins, 2019. "Piecewise constant martingales and lazy clocks," LIDAM Reprints CORE 2990, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    1. Brigo, Damiano & Jeanblanc, Monique & Vrins, Frédéric, 2020. "SDEs with uniform distributions: Peacocks, conic martingales and mean reverting uniform diffusions," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 3895-3919.

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