Laboratoire de Probabilités, Statistique et Modélisation
Présentation
Le Laboratoire de Probabilités, Statistique et Modélisation, dans sa forme actuelle, a résulté, au 1er janvier 1999, de la fusion de l'ancien Laboratoire de probabilité de l'université Paris 6 avec l'équipe de Probabilités et statistique de l'université Paris Diderot.
Le laboratoire compte environ 70 enseignants-chercheurs permanents, 50 thésards, une équipe administrative de 6 personnes. Il accueille de plus les activités de deux masters deuxième année, ce qui représente plus de 200 étudiants chaque année.
La thématique du laboratoire s'inscrit dans le domaine des mathématiques appliquées et a pour objet la modélisation, la description et l'estimation des phénomènes aléatoires. Les thèmes de recherche abordés ici concernent des domaines très variés et recouvrent aussi bien des mathématiques fondamentales que des applications dans des domaines aussi divers que la médecine, les sciences humaines, l'astrophysique, les assurances ou la finance...
Thèmes de recherche
1. Théorie ergodique et systèmes dynamiques
2. Modélisation stochastique
3. Mouvement brownien et calcul stochastique
4. Statistiques
Equipes de recherche
Le laboratoire comprend six équipes :
- Théorie ergodique et systèmes dynamiques,
- Modélisation stochastique,
- Mouvement brownien et calcul stochastique,
- Statistique,
- Probabilités numériques et mathématiques financières,
- Probabilités-statistiques-biologie.
[hal-00773708] Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps
Date: 14 1 月 2013 - 15:27
Desc: We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). We first provide general existence, uniqueness and comparison theorems for RBSDEs with jumps in the case of a RCLL adapted obstacle. We then show that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of an optimal stopping time is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, robust optimal stopping problems related to the case with model ambiguity are investigated.
[hal-01666869] CONVENIENT MULTIPLE DIRECTIONS OF STRATIFICATION
Date: 18 12 月 2017 - 17:59
Desc: This paper investigates the use of multiple directions of stratification as a variance reduction technique for Monte Carlo simulations of path-dependent options driven by Gaussian vectors. The precision of the method depends on the choice of the directions of stratification and the allocation rule within each strata. Several choices have been proposed but, even if they provide variance reduction, their implementation is computationally intensive and not applicable to realistic payoffs, in particular not to Asian options with barrier. Moreover, all these previously published methods employ orthogonal directions for multiple stratification. In this work we investigate the use of algorithms producing convenient directions, generally non-orthogonal, combining a lower computational cost with a comparable variance reduction. In addition, we study the accuracy of optimal allocation in terms of variance reduction compared to the Latin Hypercube Sampling. We consider the directions obtained by the Linear Transformation and the Principal Component Analysis. We introduce a new procedure based on the Linear Approximation of the explained variance of the payoff using the law of total variance. In addition, we exhibit a novel algorithm that permits to correctly generate normal vectors stratified along non-orthogonal directions. Finally, we illustrate the efficiency of these algorithms in the computation of the price of different path-dependent options with and without barriers in the Black-Scholes and in the Cox-Ingersoll-Ross markets.
[hal-00665852] On some expectation and derivative operators related to integral representations of random variables with respect to a PII process
Date: 2 2 月 2012 - 20:30
Desc: Given a process with independent increments $X$ (not necessarily a martingale) and a large class of square integrable r.v. $H=f(X_T)$, $f$ being the Fourier transform of a finite measure $\mu$, we provide explicit Kunita-Watanabe and Föllmer-Schweizer decompositions. The representation is expressed by means of two significant maps: the expectation and derivative operators related to the characteristics of $X$. We also provide an explicit expression for the variance optimal error when hedging the claim $H$ with underlying process $X$. Those questions are motivated by finding the solution of the celebrated problem of global and local quadratic risk minimization in mathematical finance.
[hal-01458419] Consistent Utility of Investment and Consumption : a forward/backward SPDE viewpoint *
Date: 6 2 月 2017 - 19:50
Desc: This paper provides an extension of the notion of consistent progressive utilities U to consistent progressive utilities of investment and consumption (U, V). It discusses 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 or a given set of test processes, we establish a stochastic partial differential equation (SPDE) of Hamilton-Jacobi-Bellman (HJB)-type that U has to satisfy. 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 with the HJB-SPDE two SDEs, 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.
[hal-00714354] Asymptotic expansion and central limit theorem for multiscale piecewise-deterministic Markov processes
Date: 4 7 月 2012 - 11:48
Desc: We consider a general class of piecewise-deterministic Markov processes with multiple time-scales. In line with recent results on the stochastic averaging principle for these processes, we obtain a description of their law through an asymptotic expansion. We further study the fluctuations around the averaged system in the form of a central limit theorem, and derive consequences on the law of the first passage-time. We apply the mathematical results to the Morris-Lecar model with stochastic ion channels.
Autres contacts
U.F.R. Mathématiques
Sophie-Germain
75013 PARIS