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 evenements:seminaireproba-math-fi [2019/04/01 08:09]Valérie Picot evenements:seminaireproba-math-fi [2020/05/06 11:30]Valérie Picot Both sides previous revision Previous revision 2020/05/06 11:32 Valérie Picot 2020/05/06 11:30 Valérie Picot 2020/05/06 11:30 Valérie Picot 2020/04/20 15:21 Valérie Picot 2020/04/20 15:11 Valérie Picot 2020/04/06 11:36 Valérie Picot 2020/03/12 07:25 Valérie Picot 2020/03/09 08:45 Valérie Picot 2020/02/26 19:28 Valérie Picot 2020/02/26 19:21 Valérie Picot 2020/02/09 18:52 Valérie Picot 2020/01/17 09:34 Valérie Picot 2020/01/17 09:33 Valérie Picot 2019/12/16 21:50 Valérie Picot 2019/12/16 21:50 Valérie Picot 2019/11/25 13:46 Valérie Picot 2019/11/04 13:23 Valérie Picot 2019/10/14 08:36 Valérie Picot 2019/10/02 08:15 Valérie Picot 2019/09/16 21:47 Arnaud Gloter 2019/09/12 09:58 Valérie Picot 2019/05/03 11:37 Valérie Picot 2019/04/18 08:34 Valérie Picot 2019/04/08 20:03 Valérie Picot 2019/04/01 08:09 Valérie Picot 2019/03/19 13:03 Valérie Picot 2019/02/18 10:43 Valérie Picot 2019/02/08 12:31 Valérie Picot 2019/01/31 12:45 Valérie Picot 2019/01/25 16:13 Valérie Picot 2019/01/07 09:29 Valérie Picot 2018/11/22 19:47 Valérie Picot 2018/09/26 15:10 Valérie Picot 2018/09/26 15:09 Valérie Picot 2018/09/05 14:42 Valérie Picot 2018/06/11 11:51 Valérie Picot 2018/05/14 13:18 Valérie Picot 2018/03/30 15:47 Valérie Picot 2018/03/27 12:37 Valérie Picot 2018/03/27 12:16 Valérie Picot 2018/03/27 11:53 Valérie Picot 2018/03/22 21:24 Valérie Picot 2018/03/22 21:23 Valérie Picot 2018/03/14 08:14 Valérie Picot 2018/03/01 20:48 Valérie Picot 2018/02/28 01:09 Arnaud Gloter 2018/02/28 01:07 Arnaud Gloter 2018/01/30 12:45 Valérie Picot 2018/01/30 12:08 Valérie Picot 2018/01/30 12:08 Valérie Picot Next revision Previous revision 2020/05/06 11:32 Valérie Picot 2020/05/06 11:30 Valérie Picot 2020/05/06 11:30 Valérie Picot 2020/04/20 15:21 Valérie Picot 2020/04/20 15:11 Valérie Picot 2020/04/06 11:36 Valérie Picot 2020/03/12 07:25 Valérie Picot 2020/03/09 08:45 Valérie Picot 2020/02/26 19:28 Valérie Picot 2020/02/26 19:21 Valérie Picot 2020/02/09 18:52 Valérie Picot 2020/01/17 09:34 Valérie Picot 2020/01/17 09:33 Valérie Picot 2019/12/16 21:50 Valérie Picot 2019/12/16 21:50 Valérie Picot 2019/11/25 13:46 Valérie Picot 2019/11/04 13:23 Valérie Picot 2019/10/14 08:36 Valérie Picot 2019/10/02 08:15 Valérie Picot 2019/09/16 21:47 Arnaud Gloter 2019/09/12 09:58 Valérie Picot 2019/05/03 11:37 Valérie Picot 2019/04/18 08:34 Valérie Picot 2019/04/08 20:03 Valérie Picot 2019/04/01 08:09 Valérie Picot 2019/03/19 13:03 Valérie Picot 2019/02/18 10:43 Valérie Picot 2019/02/08 12:31 Valérie Picot 2019/01/31 12:45 Valérie Picot 2019/01/25 16:13 Valérie Picot 2019/01/07 09:29 Valérie Picot 2018/11/22 19:47 Valérie Picot 2018/09/26 15:10 Valérie Picot 2018/09/26 15:09 Valérie Picot 2018/09/05 14:42 Valérie Picot 2018/06/11 11:51 Valérie Picot 2018/05/14 13:18 Valérie Picot 2018/03/30 15:47 Valérie Picot 2018/03/27 12:37 Valérie Picot 2018/03/27 12:16 Valérie Picot 2018/03/27 11:53 Valérie Picot 2018/03/22 21:24 Valérie Picot 2018/03/22 21:23 Valérie Picot 2018/03/14 08:14 Valérie Picot 2018/03/01 20:48 Valérie Picot 2018/02/28 01:09 Arnaud Gloter 2018/02/28 01:07 Arnaud Gloter 2018/01/30 12:45 Valérie Picot 2018/01/30 12:08 Valérie Picot 2018/01/30 12:08 Valérie Picot 2018/01/30 12:07 Valérie Picot Last revision Both sides next revision Line 4: Line 4: Cliquer [[:​contact|ici]] pour plus d'​informations sur les moyens d'​accès. \\ Cliquer [[:​contact|ici]] pour plus d'​informations sur les moyens d'​accès. \\ - __Contact__ : Etienne Chevalier, Dasha Loukianova, Sergio Pulido \\ + __Contact__ : Christophe Profeta, Sergio Pulido, Abass Sagna \\ **__Exposés de l'​année 2019__ :** **__Exposés de l'​année 2019__ :** + **jeudi 4 juin à 14h :** Rafael Serrano ​ (Universidad del Rosario, Colombia) //TBA. // + ++ Voir résumé |  \\ + + **jeudi 7 mai à 14h :** Marc Chataignier ​ (UEVE, LaMME, Evry) //Deep local volatility. // + ++ Voir résumé |  \\  L'​apprentissage profond est apparu comme une nouvelle façon de calculer rapidement le prix d’options notamment à des fins de calibration et d’estimation des sensibilités. Cependant, la plupart de ces approches dans la littérature ne s’assurent pas de la non-arbitrabilité des prix estimés. + + Dans cet article, nous présentons une approche d'​apprentissage profond pour l'​interpolation sans arbitrage des prix des options vanilles européennes. En particulier,​ nous détaillons les changements apportés à la méthodologie standard pour imposer des contraintes de non-arbitrage et spécifions expérimentalement les paramètres requis pour conserver une précision adéquate. Un ajout notable est l'​utilisation de la formule Dupire pour encadrer la volatilité locale associée aux prix des options (non arbitrables),​ lors de l’entraînement du réseau. + + De cette façon, nous obtenons un réseau neuronal capable d'​interpoler conjointement le prix et la volatilité locale. + ++ + **jeudi 23 avril à 14h :** ​Adrien Barrasso ​ ​ (CMAP, École Polytechnique)//​TBA. // + ++ Voir résumé |  \\ Nous commencerons par faire quelques rappels sur ce que sont les jeux à champ moyen (MFG) avec ou sans bruit commun, et les notions de solutions fortes, faibles, relâchées. Puis sur les Équations Différentielles Stochastiques Rétrogrades d'​ordre 2 (2BSDEs) qui apparaissent naturellement dans des problèmes de contrôle de volatilité et sont liées à des EDP (complètement) non-linéaires d'​ordre deux. Enfin nous présenterons un résultat d'​existence d'​équilibre pour un jeu à champ moyen avec bruit commun et contrôle de volatilité,​ ainsi qu'un théorème de représentation de cet équilibre par un 2BSDE de type McKean-Vlasov. + ++ + + **jeudi 9 avril à 15h :** Yating LIU ​ (CEREMADE, Université Paris Dauphine - PSL) //​Functional convex order for the scaled McKean-Vlasov processes. // + ++ Voir résumé |  \\ We establish the functional convex order results for two scaled McKean-Vlasov processes X = (Xt)_t∈[0,​ T] and Y = (Yt)_t∈[0,​ T] defined by dX_t=(aX_t+b)dt+sigma(t,​ X_t, mu_t)dB_t and dY_t=(aY_t+b)dt+theta(t,​ Y_t, nu_t)dB_t: if we make the convexity and monotony assumption (only) on sigma and if sigma <= theta with respect to the partial matrix order, the convex order for the initial random variable X0 <= Y0 can be propagated to the whole path of process X and Y. That is, if we consider a convex functional F defined on the path space, we have EF(X)<= EF(Y); for a convex functional G defined on the product space involving the path space and its marginal distribution space, we have EG(X, (mu_t)_t∈[0,​ T]) <= EG(Y, (nu_t)_t∈[0,​ T]) under appropriate conditions. The dual case is also valid, that is, if theta <= sigma and Y0 <= X0 with respect to the convex order, then EF(Y) <= EF(X) and EG(Y, (nu_t)_t∈[0,​ T]) <= EG(X, (mu_t)_t∈[0,​ T]). The proof is based on several forward and backward dynamic programming and the convergence of the Euler scheme of the McKean-Vlasov equation. Joint work with Gilles Pagès. + ++ + + **jeudi 26 mars à 14h :** Antonello Pesce  ​ (Università di Bologna) //​Parametrix techniques for spds. // + ++ Voir résumé |  \\ We discuss the use of the parametrix method in the context of SPDEs in Holder spaces. We address the multi-dimensional parabolic case and a two-dimensional degenerate case showing existence, regularity, Gaussian type estimates of a stochastic fundamental solution, and applications to filtering theory. Our analysis is based on a Wentzell’s reduction of the SPDE to a PDE with random coefficients and point-wise controls for related stochastic flows of diffeomorphism. + ++ + + **jeudi 19 mars à 14h :** Sarah Lemler  ​ (Centrale Supélec) //TBA. // + ++ Voir résumé |  \\ We consider a 1-dimensional diffusion process X with jumps. The particularity of this model relies in the jumps which are driven by a multidimensional Hawkes process denoted N. We consider the Markovian case for which we were able to establish ergodicity results for process X. We will present in this talk the study of a nonparametric estimator of the drift coefficient of this original process. We have constructed an estimator based on discrete observations of the process X in a high frequency framework with a large horizon time and on the observations of the process N. We have obtained adaptive results that are comparable with the one obtained in the nonparametric regression context. We have finally conducted a simulation study in which we first focus on the implementation of the process and then on showing the good behavior of the estimator. + ++ + + **jeudi 12 mars à 13h30 :** Matteo Basei  ​ (EDF)//TBA. // + ++ Voir résumé |  \\ We consider a general class of nonzero-sum stochastic games with impulse controls. By means of a suitable system of quasi-variational inequalities,​ we provide a verification theorem for the equilibrium strategies. We then present some examples and applications. Finally, we consider some extensions and future research directions. + ++ + + **jeudi 12 mars à 14h15 :** Thorsten Schmidt ​ (university of Friburg, Germany) ​ //TBA. // + ++ Voir résumé |  \\This work is an attempt to fundamentally study the valuation of insurance contracts. We start from the observation that insurance contracts are inherently linked to financial markets, be it by the link to interest rates, or -- as in hybrid products, equity-linked life insurance and variable annuities -- directly to stocks or indices. By defining trading strategies on an insurance portfolio and combining them with financial trading strategies we arrive at the notion of insurance-finance arbitrage (IFA). A fundamental theorem characterizes absence of IFA utilizing the law of large numbers and risk-neutral valuation. As a key result we obtain a simple valuation rule which excludes IFA. + + Utilizing ​ the theory of enlargements of filtrations we are able to construct a tractable framework for general valuation results. For practical applications,​ we provide an affine formulation of the driving quantities which leads to explicit valuation formulas for a large class of  hybrid products. + ++ + + **jeudi 13 février à 13h30 :** Gilles Pagès ​ ​ (Sorbonne Université,​ UPMC) //Schéma d'​Euler à pas décroissant d'une diffusion ergodique et algorithme de Langevin. // + ++ Voir résumé |  \\  Nous établissons des vitesses de convergence en variation totale et en distance de Wasserstein L^1 de la loi d'un schéma d'​Euler à pas décroissant d'une diffusion fortement ergodique vers sa loi invariante. Cela étend au cas d'un bruit multiplicatif divers résultats récents sur l'​algorithme "​Unajusted Langevin"​. ​ Nous utiliserons des estimées sous-gaussiennes de densité "à la Aronson"​ dans le cas d'un drift non bornée (à croissance linéaire). Travail joint avec F. Panloup. + ++ + + + **jeudi 23 janvier à 14h00 :** Noufel Frikha ​ (Université Paris 7) //​Well-posedness of McKean-Vlasov SDEs, related PDE on the Wasserstein space and some new quantitative estimates for propagation of chaos. // + ++ Voir résumé |  \\  In this talk, I will present some recent results on the well-posedness in the weak and strong sense of some non-linear stochastic differential equations (in the sense of McKean-Vlasov) driven by Brownian and/or jump processes which go beyond those derived from the standard Cauchy-Lipschitz theory (see e.g. the monograph of Sznitman). Then, in the Brownian setting, I will show how the underlying noise regularizes the equation and allows to prove that the transition density of the dynamics exists and is smooth, especially in the measure direction, under the uniform ellipticity assumption. Such smoothing effects then in turn allow to establish the existence and uniqueness for the Cauchy problem associated to the Kolmogorov PDE stated on the Wasserstein space with irregular terminal condition and source term. This PDE on an infinite dimensional space plays a key role in order to derive new quantitative estimates of propagation of chaos for the mean-field approximation by systems of interacting particles. + + This presentation is based on several recent works in collaboration with: P.-E. Chaudru de Raynal (Université Savoie Mont Blanc), V. Konakov (HSE Moscou), L. Li (UNSW Sydney) and S. Menozzi (Université d'Evry Val d'​Essone). + ++ + + **jeudi 19 décembre à 15h00 :** Miryana Grigorova​ (University of Leeds) //A non-linear incomplete market model with default: Pricing of European and American options// + ++ Voir résumé |  \\ We present an incomplete market model with default which consists of one risky asset with dynamics driven by two "​sources of risk", namely a Brownian motion and a compensated default martingale. Additionally to this feature, the wealth process follows non-linear dynamics with a non-linear driver f, which allows to incorporate a number of imperfections in the market. + We thus face a non-linear incomplete market with default. ​ We provide a dual formulation of the seller'​s superhedging price for a European option in terms of the supremum, over a suitable set of equivalent probability measures Q, of the non-linear f-evaluation/​expectation under Q of the payoff. ​ We also provide some related criteria for replicability of a given pay-off. ​ By a form of symmetry, we derive corresponding results for the buyer. ​ Our results rely on first establishing a non-linear optional decomposition for processes which are (non-linear) f-strong supermartingales under Q, for all Q.  This decomposition is the analogue in our framework of the well-known optional decomposition from the linear case.  We also show that the non-linear optional decomposition is equivalent to a non-linear predictable decomposition with constraints. + This result allows us to show an infinitesimal characterization of the seller'​s (superhedging) price process as the minimal supersolution of a constrained BSDE with default. + We will also discuss corresponding results for the seller'​s superhedging price of an American option. + + The talk is based on joint works with Marie-Claire Quenez and Agnès Sulem. + ++ + + **28 novembre à 14h00 :** Marie-Amélie Morlais ​(Université du Mans)  // Problème de commutation optimale avec nombre infini de modes : Une approche par “randomisation” et caractérisation par une EDSR avec contraintes sur les sauts// + ++ Voir résumé |  \\Dans une première partie de l'​exposé,​ on introduit: + + - d'une part, le problème de contrôle stochastique primal (qui n'est autre que le problème de commutation optimale avec nombre infini de modes) + -  d'​autre part, le problème de contrôle dit dual: ce dernier nécessite une construction du cadre dit "​randomisé"​ qui spécifie ​ en quoi consiste l'​ensemble des contrôles admissible dans ce nouvel espace probabilisé abstrait. + Les données du problème peuvent être à dépendance trajectorielle (en particulier,​ ceci est le cas des coefficients b et sigma définissant l’EDS associée à un processus exogène X. Les deux processus b et sigma sont contrôlés.) + Une différence majeure provient aussi du fait que l'​ensemble des modes est un espace de Borel infini éventuellement non dénombrable. + On présente les résultats majeurs du papier : + (i) l'​égalité des fonctions valeurs (associées aux problèmes primaux et duaux introduits) ; + (ii) la caractérisation de la fonction valeur commune comme la solution (minimale) d’une EDSR avec contraintes sur le terme de sauts. + Si le temps le permet, quelques idées de preuve seront données (pour le premier résultat (i)). La caractérisation à l'aide d'une Backward (avec contraintes) de la fonction valeur duale résultant d’outils relativement classiques. + Pour conclure, quelques perspectives d'​étude future seront présentées. + (Travail en commun avec Marco Fuhrman, Universita degli Studi Di Milano, Milan Italie). + ++ + + **7 novembre à 14h00 :** Lokmane Abbas Turki​(Sorbonne Universités - Paris 6)  //  Conditionnal Monte Carlo Learning for diffusions//​ + ++ Voir résumé |  \\We present a new algorithm based on One-Layered Nested Monte Carlo (1NMC) to simulate functionals $U$ of a Markov process $X$. The main originality of the proposed method comes from the fact that it provides a recipe to simulate $U_{t\geq s}$ conditionally on $X_{s}$. This recipe can be used for a large number of situations including: Backward Stochastic Differential Equations (BSDEs), Reflected BSDEs (RBSDEs), risk measures and beyond. In contrast to previous works, our contribution is based on a judicious combination between regression and 1NMC used for localization purpose. The generality, the stability and the iterative nature of this algorithm, even in high dimension, make its strength. It is of course heavier than a straight Monte Carlo (MC), however it is far more accurate to simulate quantities that are almost impossible to simulate with MC. Indeed, using the double layer of trajectories,​ we explain how to estimate and control the bias propagation. With this double layer structure, it is also possible to adjust the variance for a better description of tail events. Moreover, the parallel suitability of 1NMC makes it feasible in a reasonable computing time. This presentation explains this algorithm and details error estimates. We also provide various numerical examples with a dimension equal to 100 that are executed in few minutes on one Graphics Processing Unit (GPU). + ++ + + **17 octobre à 14h00 :** Alexandre Veretennikov ​ (University of Leeds) ​ //  On McKean-Vlasov stochastic equations// + ++ Voir résumé |  \\Weak existence will be shown for a class of McKean-Vlasov equations. Specifically results will be presented on: (a) existence for bounded Borel coefficients with non-degenerate diffusion (the class of coefficients is a bit wider than the standard linear coefficient dependence of the measure); (b) existence for unbounded Borel coefficients under linear growth given that for bounded ones existence is known; (c) existence for non-symmetric (& still non-degenerate) diffusions. In addition some results on strong existence and on weak and strong uniqueness will be stated. + ++ + + **2 octobre à 14h00 :** Sergio Pulido Nino  ​ (ENSIIE/​LaMME) ​ //  Stochastic Volterra equations// + ++ Voir résumé |  \\We obtain general weak existence and stability results for Stochastic Convolution Equations (SVEs) with jumps under mild regularity assumptions,​ allowing for non-Lipschitz coefficients and singular kernels. The motivation to study SVEs comes from the literature on rough volatility models. Our approach relies on weak convergence in Lp spaces. The main tools are new a priori estimates on Sobolev-Slobodeckij norms of the solution, as well as a novel martingale problem that is equivalent to the original equation. This leads to generic approximation and stability theorems in the spirit of classical martingale problem theory. To illustrate the applicability of our results, we consider scaling limits of nonlinear Hawkes processes and approximations of stochastic Volterra processes by Markovian semimartingales. + ++ + + **26 septembre à 14h00 :** Andrew Soane  ​ (University of Cape Town)  //  Optimal stopping with an enlarged filtration with an application to the Brownian Bridge// + ++ Voir résumé |  \\This talk will give an overview of the enlargement of filtration, focusing on the tools developed for its application,​ as well as a brief overview of optimal stopping problems from a Martingale perspective. We will then prove a relationship between the Snell envelope in the enlarged filtration and a parameterised Snell envelope in the reference filtration. Using this relationship we will then derive the optimal stopping value of a Brownian bridge, confirmed by results in the literature. + ++ + + **16 mai à 14h00 :** Aurélien ​ Alfonsi  ​ (  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.. + ++ + + **18 avril à 14h00 :** Roxana Dumitrescu  ​ (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). + ++ + + **11 avril à 14h00 :** Caroline Hillairet  ​ (ENSAE) //  Aggregation of  heterogeneous ​ consistent progressive utilities// + ++ Voir résumé |  \\We aim  to describe globally the behavior and preferences of heterogeneous agents. Our starting point is the aggregate wealth of a given economy, with a given repartition of the wealth among investors, which is not necessarily Pareto optimal. We propose a construction of an aggregate forward utility, market consistent,​that aggregates the marginal utility of the heterogeneous agents. This construction is based on the aggregation of the pricing kernels of each investor. As an application we analyze the impact of the heterogeneity and of the wealth market on the yield curve. + + Joint work with Nicole El Karoui et Mohamed Mrad. + ++ **4 avril à 14h00 :** Xiaolu Tan  ​ (Université Paris-Dauphine) //  From Martingale Optimal Transport to McKean-Vlasov Control Problems// **4 avril à 14h00 :** Xiaolu Tan  ​ (Université Paris-Dauphine) //  From Martingale Optimal Transport to McKean-Vlasov Control Problems//