Root to Kellerer

TL;DR

This paper provides a concise proof of Kellerer's Theorem, demonstrating the existence of a Markov martingale with prescribed marginals for increasing convex order distributions, using topological and coupling arguments.

Contribution

The paper offers a new, streamlined proof of Kellerer's Theorem by leveraging the topology of martingale measures and properties of Root's embedding, building on prior foundational work.

Findings

Existence of Markov martingale with given marginals for convex order increasing distributions.

Use of topological properties of martingale measures to simplify proof.

Application of Root's embedding properties in the proof.

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}.