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Conic martingales from stochastic integrals

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  • Monique Jeanblanc
  • Frédéric Vrins

Abstract

In this paper, we introduce the concept of conic martingales. This class refers to stochastic processes that have the martingale property but that evolve within given (possibly time†dependent) boundaries. We first review some results about the martingale property of solution to driftless stochastic differential equations. We then provide a simple way to construct and handle such processes. Specific attention is paid to martingales in [0, 1]. One of these martingales proves to be analytically tractable. It is shown that up to shifting and rescaling constants, it is the only martingale (with the trivial constant, Brownian motion, and geometric Brownian motion) having a separable diffusion coefficient σ(t,y)=g(t)h(y) and that can be obtained via a time†homogeneous mapping of Gaussian diffusions. The approach is exemplified by modeling stochastic conditional survival probabilities in the univariate and bivariate cases.
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Suggested Citation

  • Monique Jeanblanc & Frédéric Vrins, 2018. "Conic martingales from stochastic integrals," LIDAM Reprints CORE 2942, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvrp:2942
    Note: In : Mathematical Finance, 28, 516-535, 2018
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    References listed on IDEAS

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    1. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
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    Cited by:

    1. Cheikh Mbaye & Frédéric Vrins, 2018. "A Subordinated Cir Intensity Model With Application To Wrong-Way Risk Cva," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(07), pages 1-22, November.
    2. Cheikh Mbaye & Frédéric Vrins, 2022. "Affine term structure models: A time‐change approach with perfect fit to market curves," Mathematical Finance, Wiley Blackwell, vol. 32(2), pages 678-724, April.
    3. BRIGO, Damiano & VRINS, Frédéric, 2018. "Disentangling wrong-way risk: pricing credit valuation adjustment via change of measures," European Journal of Operational Research, Elsevier, vol. 269(3), pages 1154-1164.
    4. 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.
    5. Frédéric Vrins, 2017. "Wrong-Way Risk Cva Models With Analytical Epe Profiles Under Gaussian Exposure Dynamics," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(07), pages 1-35, November.
    6. Cheikh Mbaye & Fr'ed'eric Vrins, 2019. "An arbitrage-free conic martingale model with application to credit risk," Papers 1909.02474, arXiv.org.
    7. Damiano Brigo & Fr'ed'eric Vrins, 2016. "Disentangling wrong-way risk: pricing CVA via change of measures and drift adjustment," Papers 1611.02877, arXiv.org.

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