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 [7] by Hirsch, Profeta, Roynette and Yor (Problem 7a-7b: "Find other versions of Kellerer's Theorem"). Case b) has been answered positively by Hirsch and Roynette in [8] 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 [14] 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 (µ t) t∈T non-decreasing in the stochastic order. We conclude the paper with open problems.

On a mixture of Brenier and Strassen Theorems

“…Note that Strassen's result has been generalized at the time continuous process level by Kellerer (see also ): if a family of elements of is increasing for the convex order (a so‐called peacocks using the terminology of ), then there exists a process which is together Markovian and a martingale such that for all . See the monograph on peacocks, for the approach by Lowther and for extensions.…”

The Markov-quantile process attached to a family of marginals

“…As proved in [23,Prop. 3], Q is Markov except if a phenomenon of this type happens, see details in Example 5.11.…”

“…In most of this literature (see e.g., [14,11,9,36,24,16,19,22,39] and the references therein) the martingales may or not be Markov. The papers by Lowther [35,34] on limits of diffusion processes for the finite dimensional convergence permitted some authors to refocus on the Markov setting (see, e.g., [3,20,23]), rediscovering Kellerer's work by the way. Lowther's proof consists in adapting the local volatility coefficient of a SDE without drift-as indicated by Dupire in his very influential note [11] on financial engineering-in order to match the marginals of (µ t ) t∈R mollified in time and space.…”

Mean Residual Life Processes and Associated Submartingales

“…Hence, since σ t , t ∈ R 2 + has constant mean, σ t , t ∈ R 2 + is also ordered by the increasing convex dominance. Since Kellerer's theorem fails in the two-parameter case (see (Juillet, 2016, Theorem 2.2)), Proposition 4.3 is not sufficient for the existence of an associated submartingale to µ ε t , t ∈ R 2 + . Moreover, µ ε t , t ∈ R 2 + is not MRL ordered and the Cox-Hobson algorithm does not apply.…”