Peacocks parametrised by a partially ordered set

Abstract

We indicate some counterexamples to the peacock problem for families of (a) real measures indexed by a partially ordered set or (b) vectorial measures indexed by a totally ordered set. This is a contribution to an open problem of the book (Peacocks and Associated Martingales, with Explicit Constructions, Bocconi & Springer Series, Springer, Milan, 2011) by Hirsch et al. and Yor (Problem 7a-7b: “Find other versions of Kellerer’s Theorem”). Case (b) has been answered positively by Hirsch and Roynette (ESAIM Probab Stat 17:444-454, 2013) but the question whether a “Markovian” Kellerer Theorem hold remains open. We provide a negative answer for a stronger version: A “Lipschitz-Markovian” Kellerer Theorem will not exist. In case (a) a partial conclusion is that no Kellerer Theorem in the sense of the original paper (Kellerer, Math Ann 198:99-122, 1972) can be obtained with the mere assumption on the convex order. Nevertheless we provide a sufficient condition for having a Markovian associate martingale. The resulting process is inspired by the quantile process obtained by using the inverse cumulative distribution function of measures(µt)t∈T non-decreasing in the stochastic order. We conclude the paper with open problems.