The Markov-quantile process attached to a family of Marginals

Understand

Let $\mu$ = ($\mu$t)t$\in$R be any 1-parameter family of probability measures on R.

• Its quantile process (Gt)t$\in$R : ]0, 1[ $\rightarrow$ RR, given by Gt($\alpha$) = inf{x $\in$ R : $\mu$t(]--$\infty$, x]) > $\alpha$}, is not Markov in general.
• We modify it to build the Markov process we call "Markov-quantile".We first describe the discrete analogue: if ($\mu$n)n$\in$Z is a family of probability measures on R, a Markov process Y = (Yn)n$\in$Z such that Law(Yn) = $\mu$n 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.