Contributions to Books:
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,
We revisit Kellererīs 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 Rootīs embedding this allows for a relatively concise proof of Kellererīs 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. (Kellererīs Theorem Revisited, vol. 361, Prépublication Université dÉvry, Columbus, OH, 2012).
Electronic version of the publication:
Created from the Publication Database of the Vienna University of Technology.