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Faking Brownian motion with continuous Markov martingales

Author

Listed:
  • Mathias Beiglböck

    (University of Vienna)

  • George Lowther
  • Gudmund Pammer

    (ETH Zürich)

  • Walter Schachermayer

    (University of Vienna)

Abstract

Hamza and Klebaner (2007) [10] posed the problem of constructing martingales with one-dimensional Brownian marginals that differ from Brownian motion, so-called fake Brownian motions. Besides its theoretical appeal, this 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 and Yor (2002) [22], Hamza and Klebaner (2007) [10], Hobson (2016) [11] and Fan et al. (2015) [8], whereas continuous (but non-Markovian) fake Brownian motions were constructed by Oleszkiewicz (2008) [23], Albin (2008) [1], Baker et al. (2006) [4], Hobson (2013) [14], Jourdain and Zhou (2020) [16]. In contrast, it is known from Gyöngy (1986) [9], Dupire (1994) [7] and ultimately Lowther (2008) [17] and Lowther (2009) [20] that Brownian motion is the unique continuous strong Markov martingale with one-dimensional Brownian marginals. We took this as a challenge to construct examples of a “barely fake” Brownian motion, that is, continuous Markov martingales with one-dimensional Brownian marginals that miss out only on the strong Markov property.

Suggested Citation

  • Mathias Beiglböck & George Lowther & Gudmund Pammer & Walter Schachermayer, 2024. "Faking Brownian motion with continuous Markov martingales," Finance and Stochastics, Springer, vol. 28(1), pages 259-284, January.
    • Handle: RePEc:spr:finsto:v:28:y:2024:i:1:d:10.1007_s00780-023-00526-w
    DOI: 10.1007/s00780-023-00526-w
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    References listed on IDEAS

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    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.
    2. David Hobson & Martin Klimmek, 2015. "Robust price bounds for the forward starting straddle," Finance and Stochastics, Springer, vol. 19(1), pages 189-214, January.
    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. Peter K. Friz & Benjamin Jourdain & Thomas Wagenhofer & Alexandre Zhou, 2025. "On the Weak Error for Local Stochastic Volatility Models," Papers 2506.10817, arXiv.org.

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    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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