Lieu : Bât. I.B.G.B.I., 23 Bd. de France, à 14h00 en salle de séminaire au 3ème étage (sauf contre-indication)
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Contact : Etienne Chevalier, Dasha Loukianova, Sergio Pulido

Exposés de l'année 2019 :

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// Voir résumé \\The Martingale Optimal Transport (MOT) problem consists in maximizing a reward value among a class of martingales with given marginal distributions. It is motivated by its application in finance to obtain the no-arbitrage price bounds of derivative options in a data calibrated market. We consider a class of MOT problems and show how it could be related to a McKean-Vlasov (mean-field) control problem, which is a large population control problem. We then study the dynamic programming principle and the numerical approximation of the McKean-Vlasov control problem.

4 avril à 15h00 : Ahmed Kebaier (Université Paris 13) // Non-asymptotic error bounds for The Multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient// Voir résumé \\In this work, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We prove that, as long as the deviation is below an explicit threshold, a Gaussian-type concentration inequality optimal in terms of the variance holds for the multilevel estimator. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.

28 mars à 14h00 : Carl Graham (Ecole Polytechnique) // Théorèmes limites pour un processus de Hawkes avec auto excitation et inhibition à portée bornée// Voir résumé \\Les processus de Hawkes avec une fonction de reproduction pouvant prendre des valeurs positives et négatives permettent de modéliser des propriétés d’auto-excitation et d’auto-inhibition de leurs points. Le cas d’une fonction de reproduction positive correspond à de l'auto-excitation pure et est bien compris. Elle admet en particulier une représentation en tant que processus de branchement avec immigration qui permet d’appliquer des résultats sur les arbres de Galton-Watson. Nous utilisons des techniques de renouvellement pour obtenir des théorèmes limites dans le cas de fonctions de reproduction à support borné pouvant prendre des valeurs négatives. Nous avons en particulier obtenu des inégalités de concentration exponentielles. Une étape importante de la preuve a été de montrer l’existence de moments exponentiels pour les durées de renouvellement de files d’attente de type M/G/infini apparaissant naturellement dans ce cadre, ce qui est en soi un résultat général intéressant. Travail en collaboration avec Manon Costa, Viet Chi Tran et Laurence Marsalle

22 fevrier à 14h00 : Eva Löcherbach (Université Paris 1) // Oscillations pour des systèmes de processus de Hawkes en interactions// Voir résumé \\On considère un système de processus de Hawkes non-linéaires avec des interactions en champ moyen pour modéliser des séquences de trains de décharge dé neurones. Après avoir étudié la limite de grande population d’un tel système (autrement dit, la propriété de propagation de chaos), j’expliquerai comment les non-linéarités dans la dynamique limite engendrent du comportement périodique et donc des oscillations auto-entretenues du système. Ceci est un travail commun avec Susanne Ditlevsen..

14 fevrier à 14h00 : Tahir Chouilli (University of Alberta) // Deflators and log-optimal portfolios for markets under random horizon// Voir résumé \\The goal of this talk is to ``measure'' the impact of a random horizon on log-optimal portfolios. This random horizon is a general random time that might represent the default time of a firm, the death time of an insured, or more generally an occurrence time of an even that might impact the market somehow. Herein, in this setting, we address the num\'eraire portfolio and the log-utility maximization problem. Due to the duality between the investment strategies and the deflators, our ultimate goal translates to the description of the impact of the random horizon on the optimal deflator. Thus, our first principal result lies in explicitly describing the set of all deflators for a model stopped at a random time in different manners. Once the set of all deflators is completely and explicitly parametrized, we address the minimization problem over the set of these deflator. For the case of log utility, this optimal deflator is completely and explicitly described in different manners using the flow information generated by the initial market model only. As a result, we conclude that the random horizon leads naturally to an implied random utility. Concerning the num\'eraire portfolio, we establish a one-to-one connection between the num\'eraire portfolio for models stopped at the random time and the num\'eraire portfolio for models under ``public'' information only. This talk is based on joint works with Sina Yansori (University of Alberta).

7 fevrier à 14h00 : Yann Braouezec (IESEG School of Management) // Stress testing banks’ balance sheets : model and empirical application to the six American systemic banks// Voir résumé \\We consider a stress test model in which each bank, after an exogenous shock, may have to sell a portion of its assets in order to comply with regulatory constraints. We calibrate our model using the six banks with significant trading operations and we show that, depending on the price impact, the contagion of failures may be significant. Our results may be used to refine current stress testing frameworks by incorporating potential contagion mechanisms between banks..

31 janvier à 14h00 : Simone Scotti (Paris Diderot) // The Alpha-Heston Stochastic Volatility Model// Voir résumé \\We introduce an affine extension of the Heston model where the instantaneous variance process contains a jump part driven by alpha-stable processes with alpha in (1,2]. In this framework, we examine the implied volatility and its asymptotic behaviors for both asset and variance options. Furthermore, we examine the jump clustering phenomenon observed on the variance market and provide a jump cluster decomposition which allows to analyse the cluster processes..

24 janvier à 14h00 : Carlo Sgarra (Politecnico di Milano) // Estimation of a Self-Exciting Jump Diffusion Model for Oil Price by a Particle Markov Chain Monte Carlo Method// Voir résumé \\In this paper we propose a self-exciting jump diffusion model for oil price dynamics based on a Hawkes-type process. In particular, the jump intensity is stochastic and path dependent, implying that a jump will increase the probability of observing a new jump and this feature of the model aims at explaining the jumps clustering effect. These kind of models are now very popular in mathematical finance and financial econometrics, but the existing literature is mainly focused on the equity market. In contrast, we fit our model to spot prices related to the WTI crude oil at daily frequency and the estimation is performed by applying a suitable modification of Particle Markov Chain Monte Carlo method proposed by Andrieu \& al. \cite{holestein}. Finally, we provide an in the sample and out of the sample analysis in order to test the validity of our approach. (Paper written in Cooperation with L. Gonzato, Milano Bicocca and ESSEC). Exposés de l'année 2018 :

29 novembre 2018 à 14h00 : Ahmed Mtiraoui (IÉSEG School of Management) // Contrôle des EDSPRs Couplées// Voir résumé \\Nous étudions un problème de contrôle avec une fonctionnelle coût non linéaire dont le système contrôlé est dirigé par une EDSPR couplée. L’objective de ce travail est d’établir l’existence d’un contrôle optimal dans la classe des contrôle stricts, donc on montre que ce contrôle vérifie notre équation et qu’il minimise la fonctionnelle coût. La méthode consiste à approcher notre système par une suite de systèmes réguliers et on montre la convergence. En passant à la limite, sous des hypothèses de convexité, on obtient l’existence d’un contrôle optimal strict. on suit cette méthode théorique dans un cadre général ( n'importe quelles hypothèses d'existence et unicité) et on introduit quelques exemples.

18 octobre 2018 à 14h00 : Lakshithe Wagalath (IÉSEG School of Management) // Strategic fire sales and price-mediated contagion in the banking system (joint work with Y. Braouézec)// Voir résumé \\We consider a price-mediated contagion framework in which each bank, after an exogenous shock, may have to sell assets in order to comply with regulatory constraints. Interaction between banks takes place only through price impact. We first characterize the equilibrium of the strategic fire sales problem and define measures of contagion. We then calibrate our model to publicly-available data – the US banks that were part of the 2015 regulatory stress-tests – and quantify contagion effects. We finally show how our framework may be used to draw regulatory measures such as the systemic risk capital surcharge for large banks..

4 octobre à 14h00 : Nicolas Marie (ESME Sudria) // Sur la contrainte des équations différentielles dirigées par le mouvement brownien fractionnaire.// Voir résumé \\L’exposé portera sur la contrainte des solutions d’équations différentielles stochastiques dirigées par le mouvement brownien fractionnaire (mBf).Il existe essentiellement deux façons de contraindre la solution d’une équation différentielle stochastique à évoluer dans un sous-ensemble fermé de l’espace.La première méthode consiste à choisir le champ de vecteurs de l’équation de sorte qu’au bord de l’ensemble contrainte, le bruit devienne négligeable et qu’une force de rappel s’exerce sur la solution. Il s’agit d’une condition d’invariance. Dans une première partie, nous démontrons une condition nécessaire et suffisante d’invariance d’une partie convexe et fermée de l’espace par une équation différentielle dirigée par le mBf et prise au sens des trajectoires rugueuses. Une seconde méthode consiste à ajouter à la solution de l’équation un processus repoussant cette dernière à l’intérieur avec une force minimale chaque fois qu’elle touche le bord de l’ensemble contrainte. Il s’agit d’un problème de réflexion de Skorokhod. La seconde partie de l’exposé porte sur l’existence, l’unicité et l’approximation de la solution d’un problème de réflexion de Skorokhod associé à une équation différentielle dirigée par le mBf et à un processus de rafle de Moreau pour un ensemble contrainte convexe, compact et dépendant continûment du temps au sens de la distance de Hausdorff. Il s’agit de travaux en collaboration avec L. Coutin, P. Raynaud de Fitte et C. Castaing.

21 juin 2018 à 14h00 : Adélaïde Olivier (Université Paris-Sud, Laboratoire de Mathématiques d'Orsay) // Estimation du taux de division dans des modèles de croissance-fragmentation // Voir résumé \\Cette présentation sera centrée sur les modèles de croissance-fragmentation, pouvant servir à modéliser la croissance d’une population de cellules. D’un point de vue stochastique, nous nous intéressons à un système de particules évoluant à travers deux phénomènes. D’une part, les particules évoluent de façon déterministe (elles vieillissent, elles croissent). D’autre part, les particules se divisent au bout d'un temps aléatoire : une particule d'âge a ou de taille x se divise en deux nouvelles particules (d'âge 0, de taille initiale x/2) selon un taux de division B(.) dépendant de l'âge a ou de la taille x de la particule. Un objectif majeur est alors de reconstruire, de façon non-paramétrique, le taux de division. Pour cela, différents schémas d'observation seront envisagés : 1) l’observation des traits de toutes des cellules jusqu’à une génération fixée dans l’arbre généalogique de la population ; 2) l’observation de la prolifération des cellules en temps continu entre les instants 0 et T (induisant un phénomène de biais de sélection, absent du premier schéma, que nous mettrons en évidence).

31 mai 2018 à 14h00 : Stéfano de Marco (CMAP, Ecole Polytechnique) // VIX derivatives in rough forward variance models // Voir résumé \\Recently proposed models for the forward variance and the spot value of theSP500 stock index based on fractional Volterra processes specifically, the so called “rough Bergomi model” of Bayer,Friz, Gatheral 2016 are not able to account for smiles of options on VIX (THE MAJOR IMPLIED VOLATILITY INDEX ON THE SP500). Indeed, the VIX process induced by this model is essentially log-normal: any calibration to VIX market instruments is, the,out of reach. We will focus on the pricing and hedging of volatility derivatives, and on the appealing features that such an extended “rough” modeling framework processes in the term-structure of volatilities, and consider its calibration to the VIX market..

17 mai 2018 à 14h00 : Marek Rutkowski (The University of Sydney) // VIX derivatives in rough forward variance models // Voir résumé \\We develop a unified valuation theory that incorporates credit risk (defaults), collateralization and funding costs, by expanding the replication approach to a generality that has not yet been studied previously and reaching valuation when replication is not assumed. This unifying theoretical framework clarifies the relationship between the two valuation approaches: the adjusted cash flows approach of Brigo et al. (2016,2017) and the classical replication approach of Bielecki et al. (2015,2017). In particular, results of this work cover several previous papers in which the authors have examinedspecific replication-based models.

4 avril 2018 à 14h00 : Xu Mingyu (AMSS-Chine) // “ Superhedging with Ratio Constraint” // Voir résumé
In this talk, we study superhedging problem for proportion constraint, i.e. ratio constraint, by BSDE approach. Using the corresponding variational inequality introduced by BSDE with constraint, we discover some non-trivial options under ratio constraint. Then we apply Malliavin calculus to give sufficient and necessary conditions for existence of non-trivial options

29 mars 2018 à 14h00 : Frédéric Abergel (CentraleSupélec) // “ Modélisation et placement d'ordre optimal dans les marchés à carnets d'ordres” // Voir résumé
les stratégies d'exécution ou de tenue de marché (market making) optimales sont au coeur des problématiques concrètes des acteurs de marché. Dans cet exposé, je présenterai des travaux récents concernant, d'une part, la modélisation probabiliste des marchés à carnets d'ordres et, d'autre part, l'application du contrôle stochastique à la détermination de stratégies de market making. Ces résultats sont obtenus en collaboration avec X. Lu, C. Huré et H. Pham.

29 mars 2018 à 14h00 : Thibaut Mastrolia (Ecole Polytechnique) // “ Optimal make-take fees for market making regulation” // Voir résumé
We consider an exchange who wishes to set suitable make-take fees to attract liquidity on its platform. Using a principal-agent approach, we are able to describe in quasi- explicit form the optimal contract to propose to market makers. This contract depends essentially on the market maker inventory trajectory and on the volatility of the asset. We also provide the optimal quotes that should be displayed by the market maker. The simplicity of our formulas allows us to analyze in details the effects of optimal contracting with an exchange, compared to a situation without contract. We show in particular that it leads to a higher quality of the liquidity and lower trading costs for investors.

8 mars 2018 à 14h00 : Charlotte Dion (Sorbonne Université, LPSM) // “ Méthodes de classification multi-classes pour des trajectoires de diffusions ” // Voir résumé
Les récentes avancées technologiques ont généré un grand nombre de données qui peuvent être modélisées par des données fonctionnelles. Dans ce travail, nous nous intéressons au problème de classification supervisée multi-classe dans le cas où les variables explicatives sont définies comme des processus de diffusions. Dans ce contexte, on donne une forme explicite du classifieur de Bayes. Puis nous proposons deux procédures de classification de type plug-in qui s'appuient sur des observations discrètes et en temps fini. Nous prouvons la consistance des procédures et nous les illustrons numériquement.

15 fevrier 2018 à 14h00 : Caroline Hillairet (ENSAE) // “ Equilibrium transactions with large investors and indifferent market makers” // Voir résumé \\We provide an extension of the notion of consistent progressive utilities U to consistent progressive utilities of investment and consumption (U;V). We discuss the notion of market consistency in this forward framework, compared to the classic backward setting with a given terminal utility, and whose value function is an example of such consistent forward utility. To ensure the consistency with the market model on a given set of test processes, we establish a stochastic partial differential equation (SPDE) of Hamilton- Jacobi-Bellman (HJB)-type to be satisfied by U. This SPDE highlights the link between the utility of wealth U and the utility of consumption V, and between the drift and the volatility characteristics of the utility U. By associating two SDEs with the HJB-SPDE, we discuss the existence and the uniqueness of a concave solution. Finally, we provide explicit regularity conditions and characterize the consistent pairs of consistent utilities of investment and consumption. Some examples, such as power utilities, illustrate the theory. We give some applications on the yield curve modeling. .

8 fevrier 2018 à 14h00 : Uladzislau Stazhynski (SGCIB & Ecole polytechnique) // “ Uncertainty Quantification for Stochastic Approximation Limits” // Voir résumé \\The method of stochastic approximation (SA) is widely used in various applied domains, such as optimization, parameter estimation, adaptive control, stochastic gradient descent methods in machine learning and efficient tail computations among others.  Recently we proposed a new efficient method (called the USA algorithm) for the uncertainty quantification of the stochastic approximation limits based on chaos expansion techniques. Il allows to efficiently reconstruct the values of the SA limit for the whole range of an uncertain parameter values  within a single procedure, avoiding  costly nested calculations. We prove the almost sure convergence and, more recently, analyze the L2-convergence rate of the USA algorithm . The applications of the our method, besides model uncertainty, include sensitivity analysis and quasi-regression in the sense of reconstructing a whole unknown function, for instance in the context of nested Monte Carlo computations involving a non-linear inner function. Important applications in finance include the calculation of risk measures (such as VaR and CVaR) under uncertainty, and the calculation of XVAs.

8 fevrier 2018 à 15h00 : Michalis Anthropelos (University of Piraeus) // “ Equilibrium transactions with large investors and indifferent market makers” // Voir résumé \\We consider a market of financial securities where large investors trade with market makers at their indifference pricing. We adapt the model of Bank and Kramkov (2015) and, under a large class of utility functions, we give conditions that guarantee the existence and the uniqueness of an individual investor's optimal order. We then consider the case of two large investors who place together their orders to market makers. For this, we establish a notion of equilibrium transaction, where the investors' sharing of the aggregate order and the price are endogenously determined. The existence of such equilibrium is proved and the indicative example of exponential utility is extensively analysed. This is a joint work with P. Bank and S. Gokay (TU-Berlin). .

1 fevrier 2018 à 14h00 : Kei Noba (université de Kyoto) // “ Approximation and duality problems of refracted processes” // Voir résumé \\Using excursion theory, we construct a Markov process with no positive jumps whose positive and negative motions are given by different standard processes. The resulting process is a generalization of Kyprianou–Loeffen's refracted Lévy processes. We discuss approximation problem for our generalized refracted Lévy processes by removing small jumps and taking the limit as the removal level tends to zero. We also discuss conditions for refracted processes to have dual processes. .

1 fevrier 2018 à 15h00 : Eduardo Abi Jaber (Paris Dauphine (CEREMADE) et AXA) // “ Affine Volterra processes” // Voir résumé \\A growing body of empirical research indicates that volatility fluctuates more rapidly than Brownian motion, which is inconsistent with standard semimartingale models. Fractional volatility models and their relatives have emerged as compelling alternatives- however, their non-Markovian structure makes computations more difficult. We show that, for a large class of such models, it is nonetheless possible to compute the characteristic function by solving an integral equation similar to the Riccati equations associated with standard affine processes. Joint work with Martin Larsson and Sergio Pulido. .

25 janvier 2018 à 14h00 : Cyril GRUNSPAN (ESILV) // “ Bitcoin, sécurité et stabilité” // Voir résumé \\Le Bitcoin tient lieu du miracle cryptographique à plus d'un titre. Il constitue la première tentative réussie de monnaie numérique totalement décentralisée et matérialise le concept de contrat automatique prophétisé par Nick Szabo dès 1993. Il établit de fait un lien entre sécurité informatique et théorie des probabilités élémentaires. Il ouvre de nouvelles perspectives en finance et sur l'organisation des sociétés modernes en général. Dans une première partie, nous expliquons les fondamentaux du protole du Bitcoin. Dans une deuxième partie, nous passons en revue deux attaques théoriques (double dépense et minage égoïste) et des moyens de les contrer. .

Exposés de l'année 2017 :

9 novembre 2017 à 14h00 : Vlad Bally (Université Paris-Est Marne-la-Vallée) // “ Regularity for jump equations using an interpolation method” // Voir résumé \\We consider a sequence of Markov semigroups Pn(t) with infinitesimal op- erators Ln and we assume that Ln → L where L is the infinitesimal operator of a Markov semigroup P. The Trotter Catto theorem says that Pn → P. Our first problem is the following: we assume that Pn has the regularity property Pn(t, x, dy) = pn(t, x, y)dy and we ask under which conditions P inherits this property: P(t, x, dy) = p(t, x, y)dy. If pn → p in a suitable way, this is trivial. But our aim is to prove such a result even in the case when pn blows up. So a first result concerns an abstract criterion going in this sense. A second result concerns an application to jump type diffusions. We discuss the regularity of the semigroup asociated to a stochastic equation with jumps. If the intensity measure is absolutely continuous then one may treat this problem using Malliaivn calculus with respect to the jump amplitudes (as developed by Bismut, Leandre, Bichteler Garevereux, Jacod ….) But if the intensity measure is purely atomic, then this approach fails. J. Picard concived a Malliavin type calculus in which the differential operators are replaced by finite differences and used it in order to prove the above regularity. We give an alternative approach here: we replace the jumps which are smaller then 1 n by a Brownian motion and we denote by Pn the semigroup associated to the new equation. Then we use the classical Malliavin calculus associated to B in order to prove that Pn has a density pn. Of course we have Pn → P but, as n → ∞ the nois is vanishing so that pn blos up. However we succed to get the regularity for P. .

29 juin 2017 à 15h00 : Chao Zhou (Department of Mathematics - National University of Singapore) // “ Bank monitoring incentives under moral hazard and adverse selection” // Voir résumé \\In this paper, we extend the optimal securitization model of Possamaï and Pagès between an investor and a bank to a setting allowing both moral hazard and adverse selection. Following the recent approach to these problems of Cvitanić, Wan and Yang, we characterize explicitly and rigorously the so-called credible set of the continuation and temptation values of the banks, and obtain the value function of the investor as well as the optimal contracts through a recursive system of first-order variational inequalities with gradient constraints. We provide a detailed discussion of the properties of the optimal menu of contracts. This is a joint work with Nicolás Hernández Santibáñez and Dylan Possamaï. .

15 juin 2017 à 14h00 : Claude Martini (Zeliade Systems) // “Martingale measures with prespecified marginals: extremal points, cycles and perturbations” (joint work with Luciano Campi, LSE)“ // Voir résumé \\The characterization of the extremal points of the set of measures with two given marginals goes back to works by Denny, Letac and Mukerjee in the 70s-80s. The set of martingale measures with given marginals attracted recently a lot of people in the context of robust finance, where the model is known only through the prices of vanilla options conveyed by the market. We investigate in detail, in the discrete space case, the properties of extremal points of this set, and exhibit several classes of interest, altogether with a useful necessary conditions for a point to be extremal. We also discuss a candidate analogous of the no-cycle property in the non martingale case and its relations to perturbations. We illustrate our results in low cardinality situations.

18 mai 2017 à 14h00 : Claudia Ceci (Università degli Studi G.D'Annunzio Chieti Pescara) // “Unit-linked life insurance policies: optimal hedging in partially observable market models” // Voir résumé
In this paper we investigate the hedging problem of a unit-linked life insurance contract via the local risk-minimization approach, when the insurer has a restricted information on the market. In particular, we consider an endowment insurance contract, that is a combination of a term insurance policy and a pure endowment, whose benefits depend on the trend of a risky asset. To allow for mutual dependence between the financial and the insurance markets, we use the progressive enlargement of filtration approach. We consider a model where the stock price process dynamics depends on an exogenous unobservable stochastic factor that also influences the mortality rate of the insured. We characterize the optimal hedging strategy in terms of the integrand in the Galtchouk-Kunita-Watanabe decomposition of the insurance payment stream with respect to the minimal martingale measure and the available information flow. We provide an explicit formula by means of predictable projection of the corresponding hedging strategy under full information with respect to the natural filtration of the risky asset price and the minimal martingale measure. Finally, we discuss applications in a Markovian setting via filtering.

18 mai 2017 à 15h00 : Pietro Siorpaes (Imperial College London) // “Structure of martingale transports in finite dimensions” // Voir résumé
Martingale optimal transport is a variant of the classical optimal transport problem where a martingale constraint is imposed on the coupling. In a recent paper, Beiglböck, Nutz and Touzi show that in dimension one there is no duality gap and that the dual problem admits an optimizer. A key step towards this achievement is the characterization of the polar sets of the family of all martingale couplings. Here we aim to extend this characterization to arbitrary finite dimension through a deeper study of the convex order.

11 mai 2017 à 14h00 : Yaroslav Melnyk (EPFL) // “Principal Component Gaussian Affine Term Structure Models ” // Voir résumé \\We study Gaussian affine term structure models (GATSMs) that additionally satisfy principal components (PC) properties. The factors are assumed to be uncorrelated and the factor loadings are assumed to satisfy an orthogonality condition. We derive necessary and sufficient conditions for a GATSM to satisfy the PC properties and show that, in a class of observationally equivalent GATSMs, a representative model satisfying the PC properties can be chosen. Joint work with Damir Filipovic and Anders Trolle. In progress.

11 mai 2017 à 15h00 : Emmanuel Gobet (Ecole Polytechnique) // “MCMC design-based non-parametric regression for rare-event. Application to nested risk computations ” // Voir résumé \\We design and analyze an algorithm for estimating the mean of a function of a conditional expectation, when the outer expectation is related to a rare-event. The outer expectation is evaluated through the average along the path of an ergodic Markov chain generated by a Markov chain Monte Carlo sampler. The inner conditional expectation is computed as a non-parametric regression, using a least-squares method with a general function basis and a design given by the sampled Markov chain. We establish non asymptotic bounds for the L2-empirical risks associated to this least-squares regression; this generalizes the error bounds usually obtained in the case of i.i.d. observations. Global error bounds are also derived for the nested expectation problem. Numerical results in the context of financial risk computations illustrate the performance of the algorithms. Travail joint avec G. Fort and E. Moulines.

4 mai 2017 à 14h00 : François Delarue (Université Nice-Sophia Antipolis) // “Restauration d'unicité pour des équilibres champ moyen” // Voir résumé \\Les équilibres champ moyen décrivent des états consensuels asymptotiques au sein de grandes populations de particules contrôlées en interaction champ moyen. De façon générale, l'existence de tels équilibres peut être établie dans des cadres assez larges; en revanche, les conditions connues d'unicité sont restrictives. Dans cet exposé, je discuterai deux exemples pour lesquels l'unicité peut être rétablie en forçant la dynamique globale de la collectivité à l'aide d'un bruit commun à toutes les particules. .

4 mai 2017 à 15h15 : Enrico Priola (Torino University) // “Parabolic estimates and Poisson process” // Voir résumé \\We show how knowing Schauder or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on time variable with the same constants as in the case of the one-dimensional heat equation. The method is quite general and is based on using the Poisson stochastic process. We will also present other applications of the method. It looks like no other method is available at this time and it is a very challenging problem to find a purely analytic approach to proving such results. This is a joint work with N.V. Krylov.

27 avril 2017 à 14h00 : Chiara Benazzoli (University of Verona) // “Mean-Field games with controlled Jumps” // Voir résumé \\We study a family of mean-eld games (MFGs) with a controlled jump component. We establish existence of a solution in a relaxed version of the MFG and give conditions under which the optimal strategies are in fact Markovian. The proofs rely upon the notions of relaxed controls and martingale problems. Furthermore, we apply the general theory to a simple illiquid inter-bank market model and provide some numerical results. Keywords: mean field games, jump measures, controlled martingale problem, relaxed controls, illiquid interbank model. .

27 avril 2017 à 15h00 : Dylan Possamaï (Université Paris-Dauphine) // “Volatility demand management for electricity: a moral hazard approach” // Voir résumé
In this work, we propose a model of electricity demand management through a principal-agent problem, allowing to obtain almost explicit optimal compensations for the consumer. We then illustrate our findings through several numerical experiments, putting the emphasis on the practical implementation of the contracts. This is a joint work with René Aïd and Nizar Touzi.

20 avril 2017 à 15h00 : Scott Robertson (Boston University) // “Optimal Investment and Pricing in the Presence of Defaults” // Voir résumé
We consider the optimal investment problem when the traded asset may default, causing a jump in its price. For an investor with constant absolute risk aversion, we compute indifference prices for defaultable bonds, as well as a price for dynamic protection against default. For the latter problem, our work complements Sircar & Zariphopoulou (2007), where it is implicitly assumed the investor is protected against default. We consider a factor model where the asset's instantaneous return, variance, correlation and default intensity are driven by a time-homogenous diffusion X taking values in an arbitrary region E. We identify the certainty equivalent with a semi-linear degenerate parabolic partial differential equation with quadratic growth in both function and gradient. Under a minimal integrability assumption on the market price of risk, we show the certainty equivalent is a classical solution. In particular, our results cover when X is a one-dimensional affine diffusion and when returns, variances and default intensities are also affine. Numerical examples highlight the relationship between the factor process and both the indifference price and default insurance. Lastly, we show the insurance protection price is not the default intensity under the dual optimal measure. This is joint work with Tetsuya Ishikawa (Morgan Stanley).

9 mars 2017 à 14h00 : Sergio Pulido Nino (ENSIIE) // “Density of probability measures with the martingale representation property” // Voir résumé
Using the theory of analytic maps, we prove density results for measures satisfying a backward formulation of the martingale representation property. These results are useful to study equilibrium-based mechanisms of pricing. This is joint work with Dmitry Kramkov. |
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30 mars 2017 à 14h00 : Sofiane Saadane (Université Toulouse) // “TBA” // Voir résumé
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23 février 2017 à 14h00 : Julien Chevalier (Université Cergy) // “Modélisation de grands réseaux de neurones par processus de Hawkes” // Voir résumé
Nous nous intéresserons aux liens qui existent entre deux échelles de modélisation neurobiologique. À un niveau microscopique, l'activité électrique de chaque neurone est représentée par un processus ponctuel. À une plus grande échelle, un système d'EDP structuré en âge décrit la dynamique moyenne de ces activités. Nous montrerons que le modèle macroscopique (système d'EDP) peut se retrouver à partir d'un réseau de $n$ neurones en champ-moyen quand $n$ tend vers $+\infty$ via une ``Loi des grands nombres. De plus, les fluctuations du réseau de $n$ neurones autour du comportement limite/macroscopique sont caractérisées par un ``Théorème central limite. Cette étude finale permet la dérivation d'un système d'EDP stochastique, plus proche de la dynamique microscopique que le système d'EDP classique.

26 janvier 2017 à 14h00 : Frédéric Vrins (Université Catholique de Louvain) // “Disentangling wrong-way risk: pricing CVA via change of measures and drift adjustment (with D. Brigo)” // Voir résumé \\Abstract: A key driver of Credit Value Adjustment (CVA) is the possible dependency between exposure and counterparty credit risk, known as Wrong-Way Risk (WWR). At this time, addressing WWR in a both sound and tractable way remains challenging: arbitrage-free setups have been proposed by academic research through dynamic models but are computationally intensive and hard to use in practice. Tractable alternatives based on resampling techniques have been proposed by the industry, but they lack mathematical foundations. This probably explains why WWR is not explicitly handled in the Basel III regulatory framework in spite of its acknowledged importance. The purpose of this paper is to propose a new method consisting of an appealing compromise: we start from a stochastic intensity approach and end up with a pricing problem where WWR does not enter the picture explicitly. This result is achieved thanks to a set of changes of measure: the WWR effect is now embedded in the drift of the exposure, and this adjustment can be approximated by a deterministic function without affecting the level of accuracy typically required for CVA figures. The performances of our approach are illustrated through an extensive comparison of Expected Positive Exposure (EPE) profiles and CVA figures produced either by (i) the standard method relying on a full bivariate Monte Carlo framework and (ii) our drift-adjustment approximation. Given the uncertainty inherent to CVA, the proposed method is believed to provide a promising way to handle WWR in a sound and tractable way. Draft: https://arxiv.org/abs/1611.02877 .

26 janvier 2017 à 15h00 : Ibrahim EKREN (ETH Department of Mathematics) // ” Portfolio choice with permanent and temporary transaction costs“ // Voir résumé \\Abstract: In this talk we study the problem of optimal portfolio choice with permanent and temporary transaction costs. In a general Markovian model the objective of the agent is to maximise the discounted value of the future excess return penalised for risk. We establish an expansion of the value function of this problem when the transaction costs go to zero. We obtain a characterisation of the asymptotically optimal portfolio.This is a joint work in progress with Johannes Muhle-Karbe. .

Exposés de l'année 2016 :

8 Décembre 2016 à 14h00 : Todor Bilarev (Humboldt-Universität zu Berlin) // “Optimal liquidation if resilience of price impact is stochastic” // Voir résumé \\Abstract: We solve explicitly a two-dimensional singular control problem of finite fuel type in infinite time horizon. The problem stems from the optimal liquidation of an asset position in a financial market with multiplicative price impact with stochastic resilience. The optimal control is obtained as a diffusion process reflected at a non-constant free boundary. To solve the variational inequality and prove optimality, we show new results of independent interest on constructive approximations and Laplace transforms of the inverse local times for diffusions reflected at elastic boundaries. This is joint work with Dirk Becherer and Peter Frentrup. .

17 novembre 2016 à 14h00 : F​rançois​ Bolley (Paris 6) // “Inégalités de Sobolev logarithmiques en dimension finie” // Voir résumé \\Abstract: Les inégalités fonctionnelles (de Sobolev, Sobolev logarithmiques, etc.) permettent de préciser le comportement en temps petit et en temps grand de solutions de certaines EDP d'évolution (chaleur, Fokker-Planck, milieux poreux, etc.). Par ailleurs, Felix Otto a montré que certaines de ces équations peuvent s'interpréter comme un flot gradient dans un espace de mesures de probabilité (et les inégalités fonctionnelles associées comme des propriétés de convexité de certaines fonctionnelles). On présentera ces travaux fondateurs et certains de leurs développements, classiques ou plus récents.. .

20 octobre 2016 à 14h00 : Tepmony Sim (Télécom ParisTech) // “General-order Observation-driven Models” // Voir résumé \\Abstract: The class of observation-driven models (ODMs) includes the GARCH$(1,1)$ model as well as integer-valued time series models such as the log-linear Poisson GARCH of order $(1,1)$ and the NBIN-GARCH$(1,1)$ models. In this contribution, we treat the case of general-order ODMs in a similar fashion as the extension of the GARCH$(1,1)$ model to the GARCH$(p,q)$ model. More precisely, we establish the stationarity and the ergodicity as well as the consistency and the asymptotic normality of the maximum likelihood estimator (MLE) for the class of general-order ODMs, under conditions which are easy to verify. We illustrate these results with specific observation-driven time series, namely, the log-linear Poisson GARCH of order $(p,q)$ and the NBIN-GARCH$(p,q)$ models. An empirical study is also provided. .

13 octobre 2016 à 14h00 : S. Menozzi (UEVE) // “Estimées L^p et Problemes de Martingales pour les opérateurs de Kolmogorov stables dégénérés” //

6 octobre 2016 à 15h00 : Wissal Sabbagh (Laboratoire Manceau de Mathématiques, Université du Maine) // “Numerical Computation for Backward Doubly SDEs with random terminal time and Application for SPDEs” // Voir résumé
We are interested in solving numerically backward doubly stochastic differential equations (BDSDEs) with random terminal time τ. The main motivations are giving a probabilistic representation of the Sobolev’s solution of Dirichlet problem for semilinear SPDEs and providing the numerical scheme for such SPDEs. Thus, we study the strong approximation of this class of BDSDEs when τ is the first exit time of a forward SDE from a cylindrical domain. We use the Euler scheme and we provide bounds for the discrete-time approximation error.

9 juin 2016 à 14h00 : Paul Gassiat (CEREMADE, Université Paris Dauphine) // “Equations de Hamilton-Jacobi stochastiques : continuité par rapport au bruit et effets régularisants” // Voir résumé
Dans cet exposé, nous considérons des équations paraboliques stochastiques non linéaires de la forme $du = F(t,x,u,Du,D^2 u) dt + H(x,Du) \circ dB_t$. Dans la première partie, nous indiquerons comment on peut donner un sens à cette équation, en suivant notamment les idées introduites par Lions et Souganidis basées sur la théorie des solutions de viscosité. Dans la deuxième partie de mon exposé j'expliquerai comment dans le cas où $H(x,Du)=|Du|^2$, la solution $u$ peut avoir plus de régularité que la solution de l'équation déterministe, ce qui peut être mesuré par une paire de solutions d'EDS réfléchies. Cet exposé s'appuie sur des travaux en commun avec P. Friz, B. Gess, P.L. Lions et P. Souganidis.

19 mai 2016 à 14h00 : Zorana Grbac (LPMA – Université Paris Diderot) // “Existence of pricing measure(s) for multiple tenor interest rate markets” // Voir résumé
In the first part of the talk we provide theoretical foundations for the existence of equivalent local martingale measures in interest rate markets subject to multiple tenors. We consider a market in which products related to different tenors are traded and introduce an appropriate notion of admissible strategy, self-financing condition and free lunch in this context. We derive the related result on no free lunch and develop an affine framework for price processes and multiple tenor spreads. In the second part of the talk we present several possible extensions of the classical HJM setup to include multiple curves and the related martingale conditions, which ensure a well-defined notion of a fair price in these markets. Based on a specific interpretation of the interest rates and the implied zero-coupon bonds in a given multiple curve HJM-type model, we distinguish between two types of conditions: the ones related to actually traded interest rate products and the ones related to “fictitious” products in an enlarged market introduced only for modeling convenience. Based on joint works with Laurence Carassus and Wolfgang Runggaldier..

12 mai 2016 à 14h00 : Aline Duarte (Université de Cergy-Pontoise) // “Estimating the interaction graph of stochastic neural dynamics” // Voir résumé \\We address the question of statistical model selection for a class of stochastic models for biological neural nets. Models in this class are systems of interacting chains with memory of variable length. Each chain describes the activity of a single neuron, indicating whether it has a spike or not at a given time. For each neuron, the probability of having a spike depends on the entire time evolution of its presynaptic neurons since the last spike time of the neuron. When the neuron spikes, its potential is reset to a resting level, and all of its postsynaptic neurons receive an additional amount of potential. The relationship between a neuron and its pre- and postsynaptic neurons defines an oriented graph, the interaction graph of the model. The goal of this paper is to estimate this graph of interactions, based on an observation of the process up to time n, within a growing sequence of observation windows. We prove the consistency of this estimator and obtain explicit error bounds for the probability of wrong estimation of the graph of interactions..

14 avril 2016 à 14h00 : Radu Stoica (Universite Lille 1 - Laboratoire Paul Painleve) // “Modélisation probabiliste et inférence statistique pour l'analyse des données spatialisées” // Voir résumé
Cet exposé présente la construction d'une méthodologie pour détecter et caractériser les structures présentes dans des données spatialisées. Cette construction procède en trois étapes. Premièrement, un modèle de forme ou de structure est proposé à partir des données observées. Ensuite, une dynamique de simulation est construite en adéquation avec le modèle. Finalement, des procédures statistiques sont mises au point pour inférer les caractéristiques de la structure cachée et les paramètres du modèle. Chacune de ces étapes est attachée à un domaine particulier des probabilités et des statistiques. La modélisation repose sur des processus markovien spatiaux, comme les processus ponctuels marqués. La dynamique de simulation utilise les chaînes de Markov. L'inférence s'appuie sur l'analyse bayesienne, le recuit simulé, le maximum de vraisemblance, les tests. La synthèse de ces trois étapes se fait au confluent de trois domaines: la géométrie aléatoire, les chaînes de Markov et les statistiques appliquées. Cette synthèse nous a permis d'aborder des applications concrètes en analyse d'image, en science de l'environnement et en astronomie..

31 mars 2016 à 14h00 : Tran Viet Chi (Laboratoire Paul Painlevé, Université des Sciences et Technologies de Lille) // “Estimation non-paramétrique des indices de Sobol d'ordre 1 pour un modèle aléatoire avec une application à l'épidémiologie” // Voir résumé
Des indicateurs possibles, lorsque l'on cherche à déterminer l'influence marginale qu'ont différentes entrées $X=(X_1,\dots X_p)$ sur une sortie Y, sont les indices de Sobol. Pour $\ell \in \{1,\dots p\}$, l'indice de Sobol d'ordre 1 correspondant est : $S_\ell=\mbox{Var}(E(Y |X_\ell))/ \mbox{Var}(Y)$. En général, ces quantités ne sont pas calculables explicitement, mais on peut les estimer à partir de simulations Monte-Carlo de $(Y,X)$. Le cas $Y=f(X)$ avec $f$ une fonction déterministe est bien connu. Dans le cas aléatoire $Y=f(X,\varepsilon)$, où $\varepsilon$ représente l'aléa du modèle, des méta-modèles sont souvent utilisés pour approcher l'espérance et la variance de la réponse par des fonctions déterministes. Nous considérons ici un estimateur non-paramétrique des indices de Sobol d'ordre 1 qui ne nécessite pas de méta-modèle. L'estimateur considéré est un estimateur à ondelettes déformées (warped wavelets), et nous étudions ses propriétés d'adaptativité. En nous inspirant des méthodes de Laurent et Massart (2000), nous établissons un effet seuil (elbow effect) pour la convergence de l'écart quadratique moyen, lorsque le nombre de simulations tend vers l'infini. Une application en épidémiologie est traitée.

24 mars 2016 à 14h00 : Xiaolu Tan (Université Paris-Dauphine) // “Branching diffusion representation of semilinear PDEs and Monte Carlo approximation” // Voir résumé
We provide a representation result of parabolic semi-linear PDEs, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod (1964), Watanabe (1965) and McKean (1975), by allowing for polynomial nonlinearity in the pair (u,Du), where u is the solution of the PDE with space gradient Du. Similar to the previous literature, our result requires a non-explosion condition which restrict to “small maturity” or “small nonlinearity” of the PDE. Our main ingredient is the automatic differentiation technique as in Henry Labordere, Tan and Touzi (2015), based on the Malliavin integration by parts, which allows to account for the nonlinearities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation..

24 mars 2016 à 15h30 : Claire Lacour (Université Paris-Sud 11, Orsay) // “Estimation non paramétrique pour les chaînes de Markov cachées” // Voir résumé
Les modèles de Markov cachés sont utilisés pour modéliser des phénomènes évoluant dans le temps et provenant de populations hétérogènes, et sont appliqués en particulier en génomique et en traitement du signal. On s'intéresse ici au cas où les observations sont à espace d'états réel (au lieu de fini). On considère ainsi une chaîne de Markov cachée (X_n,Y_n) où (X_n) est une chaîne de Markov non observée à espace d'états finis, et Y_n sachant X_n=k suit une loi absolument continue, de densité f_k. Étant donné les seules observations Y_1, …, Y_n, on cherche à retrouver à la fois la dynamique de la chaîne cachée (c'est-à-dire sa matrice de transition) et les densités f_k. On présentera deux méthodes : l'une purement matricielle et spectrale, et la deuxième par contraste moindre carrés pénalisé. Finalement, on combinera ces deux méthodes pour obtenir un estimateur à la fois quasi-optimal en théorie (du point de vue minimax), et très performant en pratique.

17 mars 2016 à 14h00 : Fontana Claudio (LPMA – Université Paris Diderot) // “Optimal investment under no unbounded profit with bounded risk” // Voir résumé
We consider the problem of optimal investment with the possibility of intermediate consumption in a general semimartingale model of an incomplete market, with preferences being represented by utility stochastic fields. In this context, we show that a full duality theory can be established as long as there is no unbounded profit with bounded risk. This condition is equivalent to the existence of an equivalent local martingale deflator and is weaker than the classical notion of no free lunch with vanishing risk. In particular, we generalize several previous results on utility maximization in the absence of martingale measures. Based on joint work with N.H. Chau, A. Cosso and O. Mostovyi.

25 février 2016 à 14h00 : Giorgia Callegaro ( University of Padova) // “Utility indifference pricing and hedging for structured contracts in energy markets.” // Voir résumé
In this paper we study the pricing and hedging of structured products in energy markets, such as swing and virtual gas storage, using the exponential utility indifference pricing approach in a general incomplete multivariate market model driven by finitely many stochastic factors. The buyer of such contracts is allowed to trade in the forward market in order to hedge the risk of his position. We fully characterize the buyer's utility indifference price of a given product in terms of continuous viscosity solutions of suitable non-linear PDEs. This gives a way to identify reasonable candidates for the optimal exercise strategy for the structured product as well as for the corresponding hedging strategy. Moreover, in a model with two correlated assets, one traded and one nontraded, we obtain a representation of the price as the value function of an auxiliary simpler optimization problem under a risk neutral probability, that can be viewed as a perturbation of the minimal entropy martingale measure. Finally, numerical results are provided..

18 février 2016 à 14h00 : Sigrid Kallblad (École Polytechnique) // “Model-Independent bounds for Asian Options: A dynamic Programming Approach.” // Voir résumé
We consider the problem of finding model-independent bounds on the price of an Asian option, when the call prices at the maturity date of the option are known. Our method di ffer from most approaches to model-independent pricing in that we consider the problem as a dynamic programming problem, where the controlled process is the conditional distribution of the asset at the maturity date. By formulating the problem in this manner, we are able to determine the model-independent price through a PDE formulation. Notably, this approach does not require specific constraints on the payoff function (e.g. convexity), and would appear to be generalisable to many related problems. This is joint work with A.M.G. Cox..

28 janvier 2016 à 14h00 : Samuel Drapeau (Shanghai Jiao Tong University) // “Risk Assessment: Marginal and Systemic Dimensions.” // Voir résumé
The financial 2007/2008 crisis revealed that too few attention was paid to a sound risk and uncertainty management in particular in its systemic dimension. In this seminar, we present a general overview of risk assessment and the different goals of which (marginal and systemic risk assessment). We will introduced a risk measure designed to address the global and intrinsic risk of multidimensional interconnected system such as banks or counter-party risk in a central clearing house. The goal is two fold: on the one hand, it provides the total amount of liquidity that has to be reserved for the system to overcome financial stress situations. On the other hand, it addresses the respective amount that each member has to reserve in function of their exposure to the whole system and the systemic risk they put on the system. We finally address the quantitative aspects by presenting how these high dimensional computations can be solved efficiently..

21 janvier 2016 à 14h30 : Agathe Guilloux (LSTA) // “Apprentissage statistique pour données observationnelles longitudinales (longitudinal observational data).” // Attention horaire exceptionnel : 14h30-15h30
Voir résumé
A partir de cinq exemples de données (cliniques, issues d'un entrepôt d'hopital, d’un système de santé, d’une mutuelle santé et marketing), nous montrons les nouveaux challenges statistiques que présentent ces données, et les questions associées. Des modèles de regression dynamique en grande dimension et pour temps d'occurrence seront introduits. Je présenterai ensuite les procédures d'estimation dans ces modèles, quelques les résultats obtenus, et les aspects algorithmiques (calcul des estimateurs). Enfin je présenterai des extensions possibles des modèles présentés.

21 janvier 2016 à 13h30 : Jean-Francois JABIR ( CIMFAV, Facultad de Ingenieria, Universidad de Valparaiso.) // “Sur un modèle de Langevin avec condition de réflexion spéculaire.” // Attention horaire exceptionnel : 13h30-14h30
Voir résumé
Dans cet exposé sera abordé le problème de l'existence et de l'unicité lié à un modèle de Langevin confiné dans un domaine borné de $R^d$ et soumis à une condition de réflexion spéculaire au bord de ce domaine. De manière générale, le modèle décrit à chaque instant, la position et la vitesse d'une particule stochastique dont la dynamique en vitesse est gouvernée par un mouvement brownien et un terme de confinement assurant le confinement de la particule au sein du domaine de confinement et modélisant les interactions entre la particule et la paroi. Après un bref exposé du modèle général et des différents problèmes théoriques associés, nous présenterons un résultat d'existence et d'unicité dans le cadre d'une dynamique confinée comprenant un terme de drift singulier. Les techniques de preuve combinent à la fois l'analyse stochastique, l'analyse d'edp cinétiques et les problèmes de trace associés. Ce travail a été mené en collaboration avec Mireille Bossy, équipe TOSCA, INRIA Sophia-Antipolis Méditerranée.

14 janvier 2016 à 14h00 : Alexandros Saplaouras (Technische Universität Berlin) // “Towards the robustness property of backward stochastic differential equations with jumps.” // Voir résumé
The term robustness in the literature of backward stochastic differential equations, hereinafter BSDE, or BSDEJ in the case a jump part exists, stands for the following property: given a (suitable) approximation $\mathcal{D}^n=\left(M^n, \xi^n, f^n\right)$ of the data $\mathcal{D}=\left(M, \xi, f\right)$ of the BSDE of interest, the solution of the $\mathcal{D}^n$-BSDE converges to the solution of the $\mathcal{D}$-BSDE. Motivated by the work of P. Briand, B. Delyon and J. Memin for robustness of BSDEs driven by the Brownian motion, we want to prove that the same holds when a BSDEJ is driven by a square integrable, quasi-left-continuous martingale, let $M$. In the general framework we regard as data the sextuple $\mathcal{D}=\left(M, \mathbb{F}, T, \xi, f, C\right)$, i.e. we will work with the (general) filtration $\mathbb{F}$, the $\mathbb{F}$-stopping time $T$ and the Lebesgue-Stieltjes integrator $C$. This work is in progress. However, on the way to obtain the result, we have overcome two intermediate problems. The first is to guarantee the existence and uniqueness of solutions of BSDEJ driven by a square integrable martingale, which is not necessarily quasi-left continuous. It generalises the one given by E. Karoui and S. J. Huang, covering the quasi-left continuous case and, moreover, it enables us to treat under the same framework both continuous and discreet times BSDEJ. The latter case is referred in the literarure as BS$\Delta$EJ. The second one is the robustness of martingale representations under different filtrations, a result which generalises the one given by J. Jacod, S. M\'el\'eard and P. Protter. Once the result is obtained, the Euler scheme for the $\mathcal{D}$-BSDEJ is immediately obtained. In the case of a L\'evy martingale $M$, the robustness for discrete time approximations is given by D. Madan, M. Pistorius and M. Stadje, a result which can be seen as a special case of the general framework.

7 janvier 2016 à 14h30 : Etienne Roquain (UPMC) // “A la recherche d'éléments statistiquement significatifs avec le test multiple.” // Attention horaire exceptionnel : 14h30-15h30
Voir résumé
Chercher une aiguille dans une botte de foin est le défi quotidien posé par l'analyse statistique des données massives (en neuro-imagerie ou en génomique par exemple). A cette fin, de nombreuses stratégies statistiques ont été mises en place, souvent basées sur des modèles dits de “grande dimension”. Dans cet exposé, nous explorons la méthodologie liée au test multiple d'hypothèses, qui a rencontré un engouement particulièrement important ces dernières décennies, notamment après le fameux papier de Benjamini et Hochberg (1995). Nous débuterons par une partie non-technique qui nous permettra de nous familiariser avec le problème. Le deuxième volet de l'exposé présentera certains aspects de ma recherche dans ce domaine, en particulier pour traiter le problème délicat de la dépendance entre les tests.

10 décembre 2015 à 14h00 : Stefano Pagliarani (École Polytechnique) // “Analytical approximations of BSDEs with non-smooth driver” // Voir résumé
We provide and analyze analytical approximations of BSDEs in the limit of small non-linearity and short time, in the case of non-smooth drivers. We identify the first and the second order approximations within this asymptotics and consider two topical financial applications: the two interest rates problem and the Funding Value Adjustment. In high dimensional diffusion setting, we present numerical tests to illustrate the efficiency of the numerical schemes. Finally, we discuss the limit of this approach by assessing the possibility of higher order expansions.

Exposés de l'année 2015 :

3 décembre 2015 à 14h00 : Thibaut Mastrolia (Paris Dauphine) // “Moral hazard under Ambiguity” // Voir résumé
In this talk, we study a Principal/Agent model by adding uncertainty about the volatility of the output for both the agent and the principal. We study more precisely the impact of the “Nature” playing against the Agent and the Principal by choosing the worst possible volatility of the output. We solve the first-best and the second-best problems associated with this framework and we show that optimal contracts are in a class of contracts, similar to those obtained by Cvitanic, Possamaï and Touzi, linear with respect to the output and its quadratic variation. We compare our results with the classical Holmström and Milgrom problem. This talk is based on a joint work with Dylan Possamaï.

26 novembre 2015 à 14h00 : Nicolas Baradel (ENSAE ) // “Optimal Control of Trading Algorithms and Bayesian parameters adjustments ” // Voir résumé
We propose a general framework for the optimal control/design of trading algorithms in situations where the impact parameters are uncertain. Given a prior on the distribution of the unknown parameters, we explain how it should evolve according to the classical Bayesian rule after each sequence of trades. Taking these progressive prior-adjustments into account, we establish a dynamic programming principle for the value function associated to the optimal control of a trading algorithm. This leads to a characterization of the optimal policy through a quasi-variational parabolic equation, which can be solved numerically. Various examples of application are discussed.

19 novembre 2015 à 14h00 : Martin Larsson (ETH) // “Conditional infimum and maxima of martingales” // Voir résumé
I will discuss the possibly surprising result that the running maximum process of a positive supermartingale can be reconstructed from its global maximum. An interesting corollary is that any positive local martingale can be reconstructed from its final value and its global maximum. I will briefly touch on connections to filtration enlargement theory as well as Skorokhod embedding. Crucial for these results is the notion of conditional infimum.

12 novembre 2015 à 14h00 : Ankush Agarwal (École Polytechnique) // “Rare event simulation related to financial risks: efficient estimation and sensitivity analysis ” // Voir résumé
We develop the reversible shaking transformation methods on path space to estimate the rare event statistics arising in different financial risk settings which are embedded within a unified framework of isonormal Gaussian process. Namely, we combine splitting methods with both Interacting Particle System (IPS) technique and ergodic transformations using Parallel-One-Path (POP) estimators. We also propose an adaptive version for the POP method and prove its convergence. We demonstrate the application of our methods in various examples which cover usual semi-martingale stochastic models (not necessarily Markovian) driven by Brownian motion and, also, models driven by fractional Brownian motion (non semi-martingale) to address various financial risks. Interestingly, owing to the Gaussian process framework, our methods are also able to efficiently handle the important problem of sensitivities of rare event statistics with respect to the model parameters.

6 novembre 2015 à 14h00 : Anthony Réveillac (INSA de Toulouse) // “Sur une nouvelle caractérisation des espaces de Malliavin-Sobolev et application à l’étude des Equations Différentielles Stochastiques Rétrogrades ” // Voir résumé
Dans cet exposé nous présenterons une nouvelle caractérisation des espaces de Malliavin-Sobolev. Nous verrons que cette dernière est particulièrement utile pour démontrer le caractère Malliavin différentielle des solutions d’équations différentielles stochastiques rétrogrades. Ce exposé est basé sur des travaux en collaboration avec Peter Imkeller, Thibaut Mastrolia et Dylan Possamaï.

22 octobre 2015 à 14h00 : Abass Sagna (ENSIIE) // “Markovian and product quantization of an $\mathbb{R}^d$-valued Euler scheme of a diffusion process with an application to Heston model” // Voir résumé
We introduce a new approach to quantize an $\mathbb{R}^d$ valued Euler scheme of a diffusion process which results from the discretization of a diffusion process using the Euler scheme. This method is based on a Markovian and componentwise product quantization and allows us, from the numerical point of view, to speak of fast quantization in dimension greater than one since the product quantization of the Euler scheme of the diffusion process and its companion weights and transition probabilities may be computed instantaneously from the Newton Raphson algorithm. We show that the resulting quantization process is a Markov chain, then, we compute the associated companion weights and transition probabilities (for the quantized process and for its components) using closed formulas. From the analytical point of view, we show that the induced quantization error at the discretization step $t_k$ is a cumulation of the marginal quantization error up to time $t_k$. Numerical experiments are performed for the pricing of a European call option in a Heston model to show the performances of the method..

8 octobre 2015 à 14h00 : Umberto Cherubini (Università Di Bologna) // “Marking-to-market systemic credit risk: Application to the European banking system” // Voir résumé
We provide a tractable model for the evaluation of the probability of a systemic credit risk shock to a cluster, defined as a shock that brings about the default of all the elements at the same time, allowing for contagion, that is a dependence structure between the idiosyncratic shocks and the systemic one. We call the model Gumbel-Marshall-Olkin (GMO), because it is an extension of the Marshall-Olkin model in which the occurrence times of the unobserved shocks are linked by a a Gumbel copula. If the model is well specified, it can be used to mark-to-market and trade a credit derivative written on the systemic shock. We show that in a pure Marshall-Olkin setting, the value of this contract is proportional to the credit index (in the same meaning of iTraxx, CDX and the like) of the cluster, and contagion introduces a bias. We show how to evaluate and validate the model on bank clusters of 8 European countries throughout the crisis. Empirically, we show that the model provides a good representation for clusters of banks in Spain, Portugal, France and the Netherlands, while it fails in the other four banking systems considered (Italy, UK, Greece and particularly Germany). .

1 octobre 2015 à 14h00 : Antonis Papapantaleon (TU Berlin) // “An equilibrium model for spot and forward prices of commodities” // Voir résumé
The aim of this project is to determine the forward price of a consumption commodity via the interaction of agents in the spot and forward commodity market. We consider a market model that consists of three agents: producers of the commodity, consumers and financial investors (sometimes also called speculators). Producers produce a fixed amount of the commodity at each time point, but can choose how much they offer in the spot market and store the rest for selling at the next time period. They also have a position in forward contracts in order to hedge the commodity price uncertainty. Consumers are setting the spot price of the commodity at each time point by their demand. Finally, investors are investing in the financial markets and, in order to diversify their portfolios, also in the forward commodity market. The equilibrium prices for the commodity are the ones that clear out the spot and forward markets. We assume that producers and investors are utility maximizers and have exponential preferences, while the consumers' demand function is linear. Moreover, the exogenously priced financial market and the demand function are driven by Lévy processes. We solve the maximization problem for each agent and prove the existence of an equilibrium. This setting allows to derive explicit solutions for the equilibrium prices and to analyze the dependence of prices on the model parameters and the agent's risk aversion. This is joint work with Michail Anthropelos and Michael Kupper. Preprint ArXiv

30 juin 2015 à 14h00 : Toby Dylan Hocking (McGill University, Montréal) // “PeakSegJoint: fast supervised peak detection via joint segmentation of multiple count data samples” // Preprint ArXiv

18 juin 2015 à 14h00 : Olivier Feron (EDF-chaire FIME) // “Modeling spot, forward and option prices of several commodities in the energy market: an econometric approach” // Voir résumé
We propose a joint modeling of spot, forward and option prices of several commodities in the Energy market, using recent developments in discrete time asset pricing methods based on the notions of stochastic discount factor and of Compound Autoregressive (or affine) stochastic processes. We show that this approach provides quasi-explicit formulae for forward and spread option prices, while allowing for a large flexibility in the modeling of dynamics, spikes and seasonality, both in the historical and the risk neutral worlds. This work is a direct extension of [2] in which this approach is applied on the electricity prices. In this work we model jointly the electricity prices as well as fuel prices, taking into account the storability properties of the latter by introducing the convenience yield. The proposed model is therefore a multivariate model with discrete latent variable for the regime switching representation, and also continuous latent variable (the convenience yield). In this context we propose an iterative approach for calibrating the model and we show an illustration in the UK Market with a model on power and gas prices.
References
[1] [Gourieroux & Monfort 2007] C. Gourieroux and A. Monfort, ”Econometric specifications of stochastic discount factor models”, Journal of Econometrics, 136(2): 509–530, 2007.
[2] [Monfort & Feron 2012] A. Mofort and O. F´eron, ”Joint econometric modeling of spot electricity prices, forwards and options”, Review of derivatives research, 15(3):217– 256, 2012.
[3] [Monfort & Pegoraro] A. Monfort and F. Pegoraro, ”Switching VARMA term structure models”. Journal of Financial Econometrics, 5(1): 105–153, 2007.

11 juin 2015 à 14h00 : Jiatu Cai (École Polytechnique) // “Asymptotic replication with modified volatility and proportional transaction costs” // Voir résumé
We consider the dynamic hedging of an European option under a general local volatility model with small proportional transaction costs. Extending the approach of Leland, we introduce a class of continuous strategies of finite cost that asymptotically (super-)replicates the payoff. An associated central limit theorem of hedging error is proved. We obtain also an explicit trading strategy minimizing the asymptotic error variance..

28 mai 2015 à 14h00 : Plamen Turkedjiev (École Polytechnique) // ” Adaptive importance sampling in linear regression algorithms for BSDEs“ // Voir résumé
We introduce an importance sampling technique for backward stochastic differential equations (BSDEs) that minimizes the conditional variance occurring in linear least- squares regression algorithms. The Radon-Nikodym derivative depends on the solution of BSDE, and therefore requires a random initialization procedure to implement. We introduce novel methods to analyze the error: firstly, we need norm stability results due to the random initialization; secondly, we avoid using concentration of measure techniques. Our theoretical results are supported by numerical experiments.

21 mai 2015 à 14h00 : Dasha Loukianova (UEVE) // “pour les vrais néophytes, mesures aléatoires de Poisson” //

7 mai 2015 à 14h00 : Philip Protter (Columbia University) // ”“ The Empirical Distribution of the Lifetimes of Financial Bubbles” // Voir résumé
The mathematical modeling of financial bubbles has become a somewhat popular topic over the last 9 years. Together with Younes Kchia and Bob Jarrow, we have developed a test that can determine when a given nonnegative price process is in a bubble or not, which amounts to determining if it is a strict local martingale under a risk neutral measure, or simply a martingale. We use this idea, refine it, and apply it to a large data set of tick data for over 3000 stocks for a period of over 10 years. We determine that, subject to certain caveats, the empirical distribution of the lifetime of financial bubbles follows a generalized gamma distribution. This is the first result we know of giving the distribution of the lifetime of financial bubbles.

23 avril 2015 à 14h00 : Mihail Zervos (London School of Economics) // “Optimal execution with multiplicative price impact” //

23 avril 2015 à 15h30: Florian Maire (ECD Dublin) // “Echantillonage par chaîne de Markov adaptative utilisant une loi de mélange incrémentale” // Voir résumé
Les méthodes de Monte Carlo par chaîne de Markov (MCMC) dites adaptatives offrent un moyen relativement simple pour un utilisateur de simuler suivant une loi cible compliquée car, contrairement aux méthodes MCMC traditionnelles, le choix explicite d'un noyau transition ne lui incombe pas. Elles s'appuient en effet sur des chaînes de Markov dont le noyau de transition évolue continuellement, de façon aléatoire (et transparente pour l'utilisateur), utilisant les états précédents de la chaine pour améliorer l'exploration de l'espace d'état et accélérer la convergence de l'échantillonneur. Dans la plupart des méthodes adaptatives, les noyaux de transition sont restreints à une famille paramétrique et le mécanisme d'adaptation consiste _a optimiser ces paramètres (pour certains critères e.g taux d'acceptation, auto corrélation de la chaine, divergence par rapport _a la loi cible, etc.). Nous nous intéressons aux deux limitations suivantes : (i) le choix d'une famille paramétrique pour le noyau de transition peut contraindre l'adaptation, (ii) l'optimisation stochastique du noyau de transition peut s'avérer délicat à mettre en oeuvre et, “contrarie quelque peu l'esprit MCMC”. Nous présenterons AIMM (Adaptive Incremental Mixture MCMC), une méthode MCMC adaptive adoptant une approche différente, contournant (i) et (ii). C'est un échantilloneur qui repose sur un noyau de proposition indépendant défini comme un modèle de mélange incremental. L'idée est de réduire, progressivement et de façon non-paramétrique, la discrépance entre la loi cible et le noyau de proposition en utilisant les poids d'importances (cf Importance Sampling), en augmentant la masse de probabilité du noyau de proposition la où la discrépance est importante. Les points d'intérêt concernent : le mécanisme d'incrémentation du noyau de transition, l'ergodicité de l'algorithme AIMM et les connexions entre méthodes MCMC et méthodes particulaires. Nous illustrerons l'efficacité de AIMM sur différentes lois cibles (multimodales, _a large queue, en dimension élevée) en comparant avec d'autres méthodes MCMC adaptatives. .

16 avril 2015 à 14h00 : Ismaël Castillo (CNRS, LPMA Paris) // “Arbres de Polya et estimation bayésienne de densité” // Voir résumé
Les arbres de Polya sont une classe de mesures de probabilité aléatoires, qui a été proposée comme loi a priori, dans un cadre Bayésien, pour l’estimation de la loi d’un échantillon i.i.d. de variables aléatoires. Dans le cadre du problème de l’estimation de densité, sous certaines conditions les arbres de Polya produisent des loi a posteriori asymptotiquement consistantes au sens de la distance de Hellinger.
Dans cet exposé, après avoir présenté quelques propriétés générales des arbres de Polya, je montrerai que le résultat de consistance précédent peut être précisé dans deux directions 1) des vitesses de convergence peuvent être obtenues 2) il est possible de caractériser la forme limite de la loi a posteriori dans un sens fonctionnel. Je présenterai quelques applications à des résultats de type Donsker pour la fonction de répartition a posteriori et à l’étude de certaines fonctionnelles de la densité.

2 avril 2015 à 14h00 : Sergio Pulido (ENSIIE/Evry) // “Financial Models with Defaultable Numéraires” // Voir résumé
Financial models are studied where each asset may potentially lose value relative to any other. To this end, the paradigm of a pre-determined numéraire is abandoned in favor of a symmetrical point of view where all assets have equal priority. This approach yields novel versions of the Fundamental Theorems of Asset Pricing, which clarify and extend non-classical pricing formulas used in the financial community. Furthermore, conditioning on non-devaluation, each asset can serve as numéraire and a classical no-arbitrage condition be formulated. It is shown when and how these local conditions can be aggregated to a global no-arbitrage condition.

26 mars 2015 à 14h00 : Roxana Dumitrescu (CEREMADE, Université Paris 9 Dauphine) // “Mixed Stochastic Control/Optimal Stopping Problems with f-expectations” //

26 mars 2015 à 15h00 : Stéphane Crépey (Atelier Sauts 1) // “calcul stochastique pour PP et PPC”//

19 mars 2015 : Areski Cousin (ISFA, Université Lyon 1) // “ On Mutivariate Extensions of Value-at-risk and Conditional-Tail-Expectation ”
Voir résumé
We propose two alternative extensions of the classical univariate Value-at-Risk (VaR) and Conditional-Tail-Expectation (CTE) in a multivariate setting (see [1] and [2]). The proposed multivariate risk measures are vector-valued measures with the same dimension as the underlying portfolio of risks. As for the multivariate Value-at-Risk measures introduced in [1], the lower- orthant CTE (resp. the upper-orthant CTE) is constructed from level sets of multivariate distribution functions (resp. of multivariate survival distribution functions). Contrary to allocation measures or systemic risk measures, these measures are also suitable for multivariate risk problems where risks are heterogenous in nature and cannot be aggregated together. Several properties have been derived. In particular, we show that the proposed multivariate VaR-s and CTE-s satisfy natural extensions of the positive homogeneity property, the translation invariance property and the comonotonic additivity property. Comparison between univariate risk measures and components of multivariate VaR and CTE are provided. We also analyze how these measures are impacted by a change in marginal distributions, by a change in dependence structure and by a change in risk level. Sub-additivity of the proposed multivariate CTE-s is provided under the assumption that all components of the random vectors are independent. Illustrations are given in the class of Archimedean copulas. References [1] A. Cousin and E. Di Bernardino, On Multivariate Extensions of Value-at-Risk, Journal of Multivariate Analysis 119,32-46 (2013) [2] A. Cousin and E. Di Bernardino, On Multivariate Extensions of Conditional-Tail-Expectation, Insurance: Mathematics and Economics, 55, 272-282 (2014) .//

12 mars 2015 : Florence Merlevede (LAMA, Université Paris-Est Marne-la-Vallée) // “ Approximation forte pour des fonctionnelles additives de chaînes de Markov géométriquement ergodiques” // Attention horaire exceptionnel : 14h30-15h30
Voir résumé
Dans cet exposé, on s'intéressera à des résultats d'approximation forte de type Komlos-Major-Tusnady pour des fonctionnelles additives de chaînes de Markov. Dans le cas de fonctionnelles bornées d'une chaîne de Markov stationnaire, Harris récurrente et géométriquement ergodique, on montrera qu'il est possible d'approximer le processus des sommes partielles par un mouvement Brownien avec une erreur d'approximation en O(log n) presque sûrement. Cet exposé est issu d'un travail en commun avec Emmanuel Rio.

29 janvier 2015 : Alexandre Boumezoued (Université Pierre et Marie Curie) // “TBA” //
Voir résumé
Hawkes processes are a class of counting processes with self-exciting dynamics which are extensively used for a variety of applications. Such processes have been introduced in the recent years in finance, covering topics such as credit risk, contagion and market microstructure. The popularity of Hawkes processes seems due to its natural formulation through its shot-noise intensity, but also to its nice interpretation in terms of branching (or clustering) mechanism. The aim of this talk is to construct a birth dynamics with ages underlying the Hawkes process, which can be recovered as the population size. The population is constructed as a measure-valued process solution to stochastic equations driven by Poisson point measures. Our approach seems to reconcile the intensity process definition and the branching representation. The virtue of the population representation is to keep track of all past events through the age pyramid. As an application, we compute some distribution properties of the Hawkes process with fertility functions satisfying linear ordinary differential equations. This appears to be a natural extension of the previous studies on the Hawkes process with exponential fertility function. The last part of the talk is dedicated to wider class of Hawkes processes including both self-excited and externally excited patterns. These have gained attention in finance to model the impact of external shocks such as news or defaults. We will construct the underlying multi-type dynamics of such processes to which our methodology to compute distribution properties could be also applied.

15 janvier 2015 : Christophe Profeta (Université d'Evry) // “Persistance et enroulements du processus de Kolmogorov stable” //

Exposés de l'année 2014 :

11 décembre 2014 : Vincent Bensaye (CMAP, Ecole polytechnique) // “Branchement en environnement aléatoire et division cellulaire” //
Voir résumé
Les processus de branchement en environnement aléatoire apparaissent dans des modèles pour la division cellulaire lorsqu'on étudie un agent à l'intérieur de la cellule qui se multiplie aléatoirement (sans interaction) et que la population d'agent est partagée aléatoirement au moment de la division. Nous étudierons comment le comportement en temps long et les événements rares des processus de branchement en environnement aléatoire renseignent sur la dynamique de la population cellulaire et font apparaître différents régimes.

4 décembre 2014 : Emilio Barucci (Politecnico di Milano) // “Health insurance, portfolio choices, and retirement in incentives” //

27 novembre 2014 : Hilmar Mai (WI Berlin) “Statistical inference for Lévy-driven SDEs: drift estimation and jump filtering”
See abstract
We review recent developments in the field of statistics for Lévy-driven jump diffusion processes. In opposite to classical estimation theory for SDEs driven by Brownian motion the jump case poses several new challenges. We will discuss some techniques to tackle these challenges for estimation of drift and volatility. In order to obtain a feasible estimation problem a jump filtering step becomes necessary followed by the actual estimation of coefficient under consideration. After the general overview we will go into more details for the drift estimation problem and present efficient estimators. Finally, we discuss some numerical examples.

13 novembre 2014 : Philip Protter (Columbia University) //“Strict Local Martingales” //
See abstract
We present the case for why strict local martingales are important in several disparate areas of finance (bubbles, illusory arbitrage, stochastic volatility, NFLVR), as well as having some intrinsic interest. We will show how one can construct examples of local martingales and in particular strict local martingales with jumps, essentially at will. This involves the concept of filtration shrinkage. We will also give conditions for when the compensators of the jump times are absolutely continuous.

6 novembre 2014 : Paolo Pigato (Université Paris-Est) // “Tube estimates for Asian type stochastic differential equation” //

9 octobre 2014 : Samuel Drapeau (Humboldt-Universität zu Berlin) // “Minimal Super Solutions of BSDE: Hedging, Duality, Markov Property ”//

2 octobre 2014 : Stefan Ankirchner (Universitat Jena) // “A generalized Donsker theorem and approximating SDEs with irregular coefficient”//