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--- evenements:seminaireproba-math-fi [2019/04/08 20:03]
Valérie Picot
+++ evenements:seminaireproba-math-fi [2019/05/03 11:37] (current)
Valérie Picot
@@ Line 7: / Line 7: @@
 **__Exposés de l'année 2019__ :**
+**16 mai à 14h00 :** <color #088A85> Aurélien  Alfonsi </color>  (  Ecole des Ponts ParisTech)  //   Approximation of OT problems with marginal moments contraints (Joint work with Rafaëll Coyaud, Virginie Ehrlacher and Damiano Lombardi)//
+++ Voir résumé |  \\Optimal Transport (OT) problems arise in a wide range of applications, from physics to economics. Getting numerical approximate solution of these problems is a challenging issue of practical importance. In this work, we investigate the relaxation of the OT problem when the marginal constraints are replaced by some moment constraints. Using Tchakaloff's theorem, we show that the Moment Constrained Optimal Transport problem (MCOT) is achieved by a finite discrete measure. Interestingly, for multimarginal OT problems, the number of points weighted by this measure scales linearly with the number of marginal laws, which is encouraging to bypass the curse of dimension. This approximation method is also relevant for Martingale OT problems. We show the convergence of the MCOT problem toward the corresponding OT problem. In some fundamental cases, we obtain rates of convergence in $O(1/n)$ or $O(1/n^2)$ where $n$ is the number of moments, which illustrates the role of the moment functions. Last, we present algorithms exploiting that the MCOT is reached by a finite discrete measure and provide numerical examples of approximations..
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+**18 avril à 14h00 :** <color #088A85> Roxana Dumitrescu </color>  (King's College London)  //  Mean-field games of optimal stopping: a relaxed solution approach//
+++ Voir résumé |  \\We consider the mean-field game where each agent determines the optimal time to exit the game by solving anoptimal stopping problem with reward function depending on the density of the state processes of agents still present in thegame. We place ourselves in the framework of relaxed optimal stopping, which amounts to looking for the optimal occupationmeasure of the stopper rather than the optimal stopping time. This framework allows us to prove the existence of the relaxedNash equilibrium and the uniqueness of the associated value of the representative agent under mild assumptions. Further, weprove a rigorous relation between relaxed Nash equilibria and the notion of mixed solutions introduced in earlier works on thesubject, and provide a criterion, under which the optimal strategies are pure strategies, that is, behave in a similar way tostopping times. Finally, we present a numerical method for computing the equilibrium in the case of potential games and showits convergence (joint work with  Peter Tankov and G. Bouveret).
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 **11 avril à 14h00 :** <color #088A85> Caroline Hillairet </color>  (ENSAE) //  Aggregation of  heterogeneous  consistent progressive utilities//

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