Root to Kellerer. (English) Zbl 1370.60083
Donati-Martin, Catherine (ed.) et al., Séminaire de probabilités XLVIII. Cham: Springer (ISBN 978-3-319-44464-2/pbk; 978-3-319-44465-9/ebook). Lecture Notes in Mathematics 2168. Séminaire de Probabilités, 1-12 (2016).
Summary: We revisit Kellerer’s theorem, that is, we show that for a family of real probability distributions \((\mu_{t})_{t\in [0, 1]}\) which increases in convex order there exists a Markov martingale \((S_{t})_{t\in [0, 1]}\) s.t. \(S_{t}\sim \mu_t\).
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 H. G. Kellerer [Math. Ann. 198, 99–122 (1972; Zbl 0229.60049)], G. Lowther [Ann. Probab. 37, No. 1, 78–106 (2009; Zbl 1210.60087)], F. Hirsch et al. [Peacocks and associated martingales, with explicit constructions. New York, NY: Springer (2011; Zbl 1227.60001); “Kellerer’s theorem revisited”, Evry: Prépublication Université d’Evry vol. 361. (2012)].
For the entire collection see [Zbl 1359.60007].
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 H. G. Kellerer [Math. Ann. 198, 99–122 (1972; Zbl 0229.60049)], G. Lowther [Ann. Probab. 37, No. 1, 78–106 (2009; Zbl 1210.60087)], F. Hirsch et al. [Peacocks and associated martingales, with explicit constructions. New York, NY: Springer (2011; Zbl 1227.60001); “Kellerer’s theorem revisited”, Evry: Prépublication Université d’Evry vol. 361. (2012)].
For the entire collection see [Zbl 1359.60007].
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