Abstract
Given a family of real probability measures increasing in convex order (a peacock) we describe a systematic method to create a martingale exactly fitting the marginals at any time. The key object for our approach is the obstructed shadow of a measure in a peacock, a generalization of the (obstructed) shadow introduced in [13, 46]. As input data we take an increasing family of measures with that are submeasures of , called a parametrization of . Then, for any α we define an evolution of the measure across our peacock by setting equal to the obstructed shadow of in . We identify conditions on the parametrization such that this construction leads to a unique martingale measure π, the shadow martingale, without any assumptions on the peacock. In the case of the left-curtain parametrization we identify the shadow martingale as the unique solution to a continuous-time version of the martingale optimal transport problem.
Furthermore, our method enriches the knowledge on the Predictable Representation Property (PRP) since any shadow martingale comes with a canonical Choquet representation in extremal Markov martingales.
Funding Statement
MB and MH have been partially supported by the Vienna Science and Technology Fund (WWTF) through project VRG17-005; they are funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 –390685587, Mathematics Münster: Dynamics–Geometry–Structure.
Acknowledgments
We thank the referees for a careful reading of the manuscript and several hints to improve the presentation of our results. We thank Mathias Beiglböck for stimulating discussions at the outset of this project and Gudmund Pammer for pointing out a mistake in an earlier version of this paper.
Citation
Martin Brückerhoff. Martin Huesmann. Nicolas Juillet. "Shadow martingales – a stochastic mass transport approach to the peacock problem." Electron. J. Probab. 27 1 - 62, 2022. https://doi.org/10.1214/22-EJP846
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