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Bridging classical and martingale Schr\"odinger bridges

Author

Listed:
  • Julio Backhoff
  • Mathias Beiglbock
  • Giorgia Bifronte
  • Armand Ley

Abstract

We investigate the martingale Schr\"odinger bridge, recently introduced by Nutz and Wiesel as a distinguished martingale transport plan between two probability measures in convex order. We show that this construction extends naturally to arbitrary dimension and admits several equivalent characterizations. In particular, we identify its continuous-time counterpart as the continuous martingale with prescribed marginals that minimizes a weighted quadratic energy measuring the deviation from Brownian motion. In the irreducible case, we prove that this continuous martingale Schr\"odinger bridge coincides with the F\"ollmer martingale, that is, with the Doob martingale associated to a suitable F\"ollmer process. More generally, we relate the martingale Schr\"odinger bridge to a variational problem over base measures and to the dual formulation of the corresponding weak optimal transport problem, thereby clarifying its connection with the classical Schr\"odinger bridge.

Suggested Citation

  • Julio Backhoff & Mathias Beiglbock & Giorgia Bifronte & Armand Ley, 2026. "Bridging classical and martingale Schr\"odinger bridges," Papers 2604.01299, arXiv.org.
    • Handle: RePEc:arx:papers:2604.01299
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    References listed on IDEAS

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    3. Stefano De Marco & Huy^en Pham & Davide Zanni, 2026. "Schr\"odinger bridges with jumps for time series generation," Papers 2602.20011, arXiv.org.
    4. Pierre Henry-Labordere, 2019. "From (Martingale) Schrodinger bridges to a new class of Stochastic Volatility Models," Working Papers hal-02090807, HAL.
    5. Julien Guyon & Romain Menegaux & Marcel Nutz, 2016. "Bounds for VIX Futures given S&P 500 Smiles," Papers 1609.05832, arXiv.org, revised Jun 2017.
    6. Julien Guyon & Romain Menegaux & Marcel Nutz, 2017. "Bounds for VIX futures given S&P 500 smiles," Finance and Stochastics, Springer, vol. 21(3), pages 593-630, July.
    7. Pierre Henry-Labordere, 2019. "From (Martingale) Schrodinger bridges to a new class of Stochastic Volatility Models," Papers 1904.04554, arXiv.org.
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