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arXiv · 1507.07690

Root to Kellerer

Mathias Beiglböck·Martin Huesmann·Florian Stebegg
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Abstract

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 Kellerer \cite{Ke72}, Lowther \cite{Lo07}, and Hirsch-Roynette-Profeta-Yor \cite{HiPr11,HiRo12}.

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Mathias Beiglböck, Martin Huesmann, Florian Stebegg. 2015-07-28. Root to Kellerer. https://doi.org/10.1007/978-3-319-44465-9_1

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