Root to Kellerer
arXiv:1507.07690 · doi:10.1007/978-3-319-44465-9_1
Abstract
We revisit Kellerer's Theorem, that is, we show that for a family of real probability distributions $(μ_t)_{t\in [0,1]}$ which increases in convex order there exists a Markov martingale $(S_t)_{t\in[0,1]}$ s.t.\ $S_t\sim μ_t$. 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 \cite{Ke72}, Lowther \cite{Lo07}, and Hirsch-Roynette-Profeta-Yor \cite{HiPr11,HiRo12}.
8 pages, 1 figure