Nicolas Juillet (University of Strasbourg): Markovinication of the quantile process
Kellerer’s theorem (1972) states that, for measures $(mu_t)_t$ in (increasing) convex order, there exists a Markov martingale, respectively a Markov submartingale, with marginals $mu_t$. In view of the Doob–Meyer decomposition theorem it may appear surprising that the corresponding statement for the stochastic order and Markov increasing processes has never been established. Note that, due to the atomic part of the measures $mu_t$, the problem can not trivially be solved by using cumulative distribution functions. Also Kellerer’s proof can not be readily adapted. That the statement is yet true without further assumption is the result of a joint work with Charles Boubel, that I will present in this talk. In particular, I will explain how Kellerer’s proof can be modified as less as possible to obtain a common proof for all three statements on Markovian martingales, submartingales and increasing processes. If times permits, I will distinguish a special process and stress it connections with optimal transport theory and the continuity equation.