Root to Kellerer

We revisit Kellerer's Theorem, that is, we show that for a family of real probability distributions (mu(t))(t is an element of[0,1]) which increases in convex order there exists a Markov martingale. (S-t)(t is an element of[0,1]) s. t. S-t similar to 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 Kellerer (Math Ann 198: 99-122, 1972), Lowther (Limits of one dimensional diffusions. ArXiv eprints, 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, Prepublication Universite dEvry, Columbus, OH, 2012).