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Faking Brownian motion with continuous Markov martingales

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  • Mathias Beiglbock
  • George Lowther
  • Gudmund Pammer
  • Walter Schachermayer

Abstract

Hamza-Klebaner posed the problem of constructing martingales with Brownian marginals that differ from Brownian motion, so called fake Brownian motions. Besides its theoretical appeal, the problem represents the quintessential version of the ubiquitous fitting problem in mathematical finance where the task is to construct martingales that satisfy marginal constraints imposed by market data. Non-continuous solutions to this challenge were given by Madan-Yor, Hamza-Klebaner, Hobson, and Fan-Hamza-Klebaner whereas continuous (but non-Markovian) fake Brownian motions were constructed by Oleszkiewicz, Albin, Baker-Donati-Yor, Hobson, Jourdain-Zhou. In contrast it is known from Gy\"ongy, Dupire, and ultimately Lowther that Brownian motion is the unique continuous strong Markov martingale with Brownian marginals. We took this as a challenge to construct examples of a "very fake'' Brownian motion, that is, continuous Markov martingales with Brownian marginals that miss out only on the strong Markov property.

Suggested Citation

  • Mathias Beiglbock & George Lowther & Gudmund Pammer & Walter Schachermayer, 2021. "Faking Brownian motion with continuous Markov martingales," Papers 2109.12927, arXiv.org.
    • Handle: RePEc:arx:papers:2109.12927
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    References listed on IDEAS

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    1. Albin, J.M.P., 2008. "A continuous non-Brownian motion martingale with Brownian motion marginal distributions," Statistics & Probability Letters, Elsevier, vol. 78(6), pages 682-686, April.
    2. Kais Hamza & Fima C. Klebaner, 2007. "A Family of Non-Gaussian Martingales with Gaussian Marginals," International Journal of Stochastic Analysis, Hindawi, vol. 2007, pages 1-19, August.
    3. Benjamin Jourdain & Alexandre Zhou, 2020. "Existence of a calibrated regime switching local volatility model," Mathematical Finance, Wiley Blackwell, vol. 30(2), pages 501-546, April.
    4. Oleszkiewicz, Krzysztof, 2008. "On fake Brownian motions," Statistics & Probability Letters, Elsevier, vol. 78(11), pages 1251-1254, August.
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    Cited by:

    1. Mathias Beiglböck & Gudmund Pammer & Walter Schachermayer, 2022. "From Bachelier to Dupire via optimal transport," Finance and Stochastics, Springer, vol. 26(1), pages 59-84, January.

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