Abstract
Let µ = (µt)t∈R be any 1-parameter family of probability measures on R. Its quantile process (Gt)t∈R : ]0, 1[ → RR, given by Gt(α) = inf{x ∈ R : µt(]−∞, x]) > α}, is not Markov in general. We modify it to build the Markov process we call “Markov-quantile”. We first describe the discrete analogue: if (µn)n∈Z is a family of probability measures on R, a Markov process Y = (Yn)n∈Z such that Law(Yn) = µn is given by the data of its couplings from n to n + 1, i.e. Law((Yn, Yn+1)), and the process Y is the inhomogeneous Markov chain having those couplings as transitions. Therefore, there is a canonical Markov process with marginals µn and as similar as possible to the quantile process: the chain whose transitions are the quantile couplings. We show that an analogous process exists for a continuous parameter t: there is a unique Markov process X with the measures µt as marginals, and being a limit for the finite dimensional topology of quantile processes where the past is made independent of the future at finitely many times (many non-Markovian limits exist in general). The striking fact is that the construction requires no regularity for the family µ. We rely on order arguments, which seems to be completely new for the purpose. We also prove new results the Markov-quantile process yields in two contemporary frameworks: – In case µ is increasing for the stochastic order, X has increasing trajectories. This is an analogue of a result of Kellerer dealing with the convex order, peacocks and martingales. Modifiying Kellerer’s proof, we also prove simultaneously his result and ours in this case. – If µ is absolutely continuous in Wasserstein space P2(R) then X is solution of a Benamou–Brenier transport problem with marginals µt. It provides a Markov probabilistic representation of the continuity equation, unique in a certain sense.
Highlights
Rather surprisingly, the basic question of the existence of a Markov martingale with prescribed marginalst∈R remains open since the partial result [28] of 1972
We introduce the following notion of a measure in Marg((μt)t∈R) “turned into a Markov law at a finite set of instants”, denoted in a way that is consistent with the notation of Theorem A(iv)
Organization of the paper. — In Section 2 we introduce in Section 2.1 kernels and transport plans, their composition and concatenation, and the Markov property expressed in this language
Summary
The basic question of the existence of a Markov martingale with prescribed marginals (μt)t∈R remains open since the partial result [28] of 1972. We introduce the following notion of a measure in Marg((μt)t∈R) “turned into a Markov law at a finite set of instants”, denoted in a way that is consistent with the notation of Theorem A(iv). See [37, Lem. 5.3] for a different statement Another consequence of this non stability of the Markov property is that it is not possible to consider the sequence of quantile processes for mollified curves μ(n) = (μt ∗ θn)t, relying on the fact that all the measures μ(tn) are diffuse, so that each Q ∈ Marg((μ(tn))t∈R) is Markov. If we call them “non decreasing”, the standard terminology when the order is total, For: process and marginal (law) canonical process, coupling & transport (plan) martingale, submartingale & increasing couplings atomic & essential atomic times increasing kernel Lipschitz kernel quantile coupling & quantile measure atomic levels a process “M made Markov at the points of R”. Acknowledgements. — The authors wish to thank Jiří Černý, Martin Huesmann, Christian Léonard, Emmanuel Opshtein and Xiaolu Tan for bibliographic or editorial suggestions as well as Michel Émery and Erwan Hillion for discussions on examples related to this work
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