By Catherine Donati-Martin, Antoine Lejay, Alain Rouault

In addition to its additional exploration of the topic of peacocks, brought in contemporary Séminaires de Probabilités, this quantity keeps the sequence’ concentrate on present learn subject matters in conventional themes reminiscent of stochastic calculus, filtrations and random matrices. additionally incorporated are a few fairly fascinating articles regarding harmonic measures, random fields and loop soups. The featured members are Mathias Beiglböck, Martin Huesmann and Florian Stebegg, Nicolas Juillet, Gilles Pags, Dai Taguchi, Alexis Devulder, Mátyás Barczy and Peter Kern, I. Bailleul, Jürgen Angst and Camille Tardif, Nicolas Privault, Anita Behme, Alexander Lindner and Makoto Maejima, Cédric Lecouvey and Kilian Raschel, Christophe Profeta and Thomas Simon, O. Khorunzhiy and Songzi Li, Franck Maunoury, Stéphane Laurent, Anna Aksamit and Libo Li, David Applebaum, and Wendelin Werner.

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2 Brownian Martingale Diffusion The main result of this section is the theorem below which show that martingale Brownian diffusions satisfy a functional convex order principle. Theorem 1 Let ,  W Œ0; T R ! R be two continuous functions with linear growth in x uniformly in t 2 Œ0; T. Let X .  / be two Brownian martingale diffusions, supposed to be the unique weak solutions starting from x at time 0, to the 42 G. Pagès stochastic differential equations with zero drift . / dXt . / . / . t; Xt /dWt ; X0 D x; (4) t 2 Œ0; T, respectively, where W .

2. q0 / so that the criterion is satisfied in (1).  1;1Ct C  1;2Ct / for T D Œ0; 1. q0 / D 1.  1 t;1 C  1 t;2 / for T D Œ0; 1. q0 / D 2g. Proof (Proof of Theorem 5) The proof is similar to the one of Proposition 3 even if more technical. In particular even if t is continuous, the value of Xt does not uniquely determine a trajectory. Nevertheless the law of the random trajectory is uniquely determined because it only depends on q and Xt . In fact, the quantile q is a function of Xt so that as described in Sect.

Of course the first and the third question have likely the same answer, yes or no. In the case T D Œ0; 1 the second question suggests an approach for the first question. Recall that it is wrong that limit of Markovian processes are Markovian. References 1. M. Beiglböck, N. Juillet, On a problem of optimal transport under marginal martingale constraints. Ann. Probab. 44(1), 42–106 (2016) 2. M. Beiglböck, M. Huesmann, F. Stebegg, Root to Kellerer, in Séminaire de Probabilités XLVIII, ed. by C. Donati-Martin, A.

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