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M. Beiglböck, M. Huesmann, F. Stebegg:
"Root to Kellerer";
in: "Séminaire de Probabilités XLVIII", Volume 2168 of the series Lecture Notes in Mathematics; C. Donati-Martin, A. Lejay, A. Rouault (ed.); Springer International Publishing, 2016, ISBN: 978-3-319-44465-9, 1 - 12.

We revisit Kellerer´s Theorem, that is, we show that for a family of real probability distributions (μ t ) t ∈ [0, 1] which increases in convex order there exists a Markov martingale (S t ) t ∈ [0, 1] s.t. S t  ∼ μ t .

To establish the result, we observe that the set of martingale measures with given marginals carries a natural compact Polish topology. Based on a particular property of the martingale coupling associated to Root´s embedding this allows for a relatively concise proof of Kellerer´s theorem.

We emphasize that many of our arguments are borrowed from Kellerer (Math Ann 198:99-122, 1972), Lowther (Limits of one dimensional diffusions. ArXiv e-prints, 2007), Hirsch-Roynette-Profeta-Yor (Peacocks and Associated Martingales, with Explicit Constructions. Bocconi & Springer Series, vol. 3, Springer, Milan; Bocconi University Press, Milan, 2011), and Hirsch et al. (Kellerer´s Theorem Revisited, vol. 361, Prépublication Université dÉvry, Columbus, OH, 2012).

http://link.springer.com/chapter/10.1007%2F978-3-319-44465-9_1#page-1