The Markov-quantile process attached to a family of marginals

Boubel, Charles; Juillet, Nicolas

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1-62.

Soit $\mu ={\left({\mu }_t\right)}_{t\in ℝ}$ une famille à un paramètre de mesures de probabilité sur $ℝ$. Son processus quantile ${\left(G_t\right)}_{t\in ℝ}:]0,1[\to {ℝ}^{ℝ}$, donné par $G_t\left(\alpha \right)=inf\{x\in ℝ:{\mu }_t\left(]-\infty ,x]\right)⩾\alpha \}$, n’est en général pas markovien. Nous le modifions pour construire le processus markovien que nous nommons « Markov-quantile » : il existe un unique processus markovien $X$ de marges ${\mu }_t$ qui est limite, pour la convergence fini-dimensionnelle, de processus quantiles dont le passé est rendu indépendant du futur en un nombre fini d’instants (beaucoup de limites non markoviennes existent en général). Il est frappant qu’aucune hypothèse de régularité sur la famille $\mu$ n’est nécessaire. En outre, si $\mu$ est croissante pour l’ordre stochastique, les trajectoires de $X$ sont croissantes. Ceci est un analogue d’un résultat de Kellerer traitant de l’ordre convexe. Dans un article associé [8] on montre aussi que si $\mu$ est absolument continue dans l’espace de Wasserstein ${𝒫}_2\left(ℝ\right)$, $X$ est solution d’un problème de transport de Benamou–Brenier avec marges ${\mu }_t$, et fournit donc une représentation markovienne de l’équation de continuité, unique dans le sens donné plus haut.

Let $\mu ={\left({\mu }_t\right)}_{t\in ℝ}$ be any $1$-parameter family of probability measures on $ℝ$. Its quantile process ${\left(G_t\right)}_{t\in ℝ}:]0,1[\to {ℝ}^{ℝ}$, given by $G_t\left(\alpha \right)=inf\{x\in ℝ:{\mu }_t\left(]-\infty ,x]\right)⩾\alpha \}$, is not Markov in general. We modify it to build the Markov process we call “Markov-quantile”: there is a unique Markov process $X$ with marginals ${\mu }_t$, 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). Strikingly, no regularity is required for the family $\mu$. Moreover, if $\mu$ is increasing for the stochastic order, $X$ has increasing trajectories. This is an analogue of a result of Kellerer dealing with the convex order. In a companion paper [8] it is also proved that if $\mu$ is absolutely continuous in the Wasserstein space ${𝒫}_2\left(ℝ\right)$, $X$ is solution of a Benamou–Brenier transport problem with marginals ${\mu }_t$, providing a Markov representation of the continuity equation, unique in the sense above.

Classification : 60A10, 28A33, 60J25, 35Q35, 60G44, 49J55
Keywords: Markov process, quantile process, optimal transport, continuity equation, increasing process, Kellerer’s theorem, martingale optimal transport, peacocks, copula
Mot clés : Processus markovien, processus quantile, transport optimal, équation de continuité, processus croissant, théorème de Kellerer, transport optimal martingale optimal, peacocks, copule

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