Search

This volume contains lecture notes from the courses given by Vlad Bally and Rama Cont at the Barcelona Summer School on Stochastic Analysis (July 2012). Rama Cont's notes provide an introduction to the Functional Itô Calculus, a non-anticipative functional calculus that extends the classical Itô calculus to path-dependent functionals of stochastic processes. This calculus leads to a new class of path-dependent partial differential equations, termed Functional Kolmogorov Equations, which arise in the study of martingales and forward-backward stochastic differential equations.

Directed polymers in random environment can be thought of as a model of statistical mechanics in which paths of stochastic processes interact with a quenched disorder (impurities), depending on both time and space. We review here main results which have been obtained during the last fifteen years, with proofs to most of the results. The material covers the diffusive behavior of the polymers in weak disorder phase studied by J. Imbrie, T. Spencer, E. Bolthausen, R. Song and X. Y. Zhou [11, 3, 25], and localization of the paths in strong disordered phase recently obtained by P. Carmona, Y. Hu, and the authors of the present article [4, 5].

We propose a general framework for modeling multiple yield curves which have emerged after the last financial crisis. In a general semimartingale setting, we provide an HJM approach to model the term structure of multiplicative spreads between FRA rates and simply compounded OIS risk-free forward rates. We derive an HJM drift and consistency condition ensuring absence of arbitrage and, in addition, we show how to construct models such that multiplicative spreads are greater than one and ordered with respect to the tenor's length. When the driving semimartingale is specified as an affine process, we obtain a flexible Markovian structure. Finally, we show that the proposed framework allows to unify and extend several recent approaches to multiple yield curve modeling., Comment: (40 pages, 4 figures)

We consider a dynamic metapopulation involving one large population of size N surrounded by colonies of size εNN, usually called peripheral isolates in ecology, where N → ∞ and εN → 0 in such a way that εNN → ∞. The main population, as well as the colonies, independently send propagules to found new colonies (emigration), and each colony independently, eventually merges with the main population (fusion). Our aim is to study the genealogical history of a finite number of lineages sampled at stationarity in such a metapopulation. We make assumptions on model parameters ensuring that the total outer population has size of the order of N and that each colony has a lifetime of the same order. We prove that under these assumptions, the scaling limit of the genealogical process of a finite sample is a censored coalescent where each lineage can be in one of two states: an inner lineage (belonging to the main population) or an outer lineage (belonging to some peripheral isolate). Lineages change state at constant rate and (only) inner lineages coalesce at constant rate per pair. This two-state censored coalescent is also shown to converge weakly, as the landscape dynamics accelerate, to a time-changed Kingman coalescent.

We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barab��si-Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel \& R��cz concerning the influence of the seed graph on the asymptotic behavior of such trees. Separately we study the geometric structure of nodes of large degrees in a plane version of Barab��si-Albert trees via their associated looptrees. As the number of nodes grows, we show that these looptrees, appropriately rescaled, converge in the Gromov-Hausdorff sense towards a random compact metric space which we call the Brownian looptree. The latter is constructed as a quotient space of Aldous' Brownian Continuum Random Tree and is shown to have almost sure Hausdorff dimension $2$., 32 pages, 11 figures

International audience; We first study an optimal stopping problem in which a player (an agent) uses a discrete stopping time in order to stop optimally a payoff process whose risk is evaluated by a (non-linear) $g$-expectation. We then consider a non-zero-sum game on discrete stopping times with two agents who aim at minimizing their respective risks. The payoffs of the agents are assessed by g-expectations (with possibly different drivers for the different players). By using the results of the first part, combined with some ideas of S. Hamadène and J. Zhang, we construct a Nash equilibrium point of this game by a recursive procedure. Our results are obtained in the case of a standard Lipschitz driver $g$ without any additional assumption on the driver besides that ensuring the monotonicity of the corresponding $g$-expectation.

A selection procedure with no optimization step called LOLA, for Learning Out of Leaders with Adaptation is proposed. LOLA is an auto-driven algorithm with two thresholding steps. The consistency of the LOL procedure (the non adaptive version of LOLA) is proved under sparsity conditions and simulations are conducted to illustrate the practical good performances of LOLA. The behavior of the algorithm is studied when instrumental variables are artificially added without a priori significant connection to the model. Finally, the problem of empirically verifying the conditional convergence hypothesis used in economics concerning the growth rate is studied. To avoid unnecessary discussion about the choice and the pertinence of instrumental variables, the model is embedded in a very high dimensional setting. Using the LOLA algorithm, a solution for modeling the link between the growth rate and the initial level of the gross domestic product is provided and the convergence hypothesis is empirically proved.

International audience; The recent financial crisis has led to so-called multi-curve models for the term structure. Here we study a multi-curve extension of short rate models where, in addition to the short rate itself, we introduce short rate spreads. In particular, we consider a Gaussian factor model where the short rate and the spreads are second order polynomials of Gaussian factor processes. This leads to an exponentially quadratic model class that is less well known than the exponentially affine class. In the latter class the factors enter linearly and for positivity one considers square root factor processes. While the square root factors in the affine class have more involved distributions, in the quadratic class the factors remain Gaussian and this leads to various advantages, in particular for derivative pricing. After some preliminaries on martingale modeling in the multi-curve setup, we concentrate on pricing of linear and optional derivatives. For linear derivatives, we exhibit an adjustment factor that allows one to pass from pre-crisis single curve values to the corresponding post-crisis multi-curve values

International audience; In this paper we consider the pricing of options on interest rates such as caplets and swaptions in the Lévy Libor model developed by Eberlein and Özkan (Financ. Stochast. 9:327-348 (2005) [9]). This model is an extension to Lévy driving processes of the classical log-normal Libor market model (LMM) driven by a Brownian motion. Option pricing is significantly less tractable in this model than in the LMM due to the appearance of stochastic terms in the jump part of the driving process when performing the measure changes which are standard in pricing of interest rate derivatives. To obtain explicit approximation for option prices, we propose to treat a given Lévy Libor model as a suitable perturbation of the log-normal LMM. The method is inspired by recent works by Černý, Denkl, and Kallsen (Preprint (2013) [6]) and Ménassé and Tankov (Preprint (2015) [14]). The approximate option prices in the Lévy Libor model are given as the corresponding LMM prices plus correction terms which depend on the characteristics of the underlying Lévy process and some additional terms obtained from the LMM model.

We study the array of point-to-point distances in random regular graphs equipped with exponential edge lengths. We consider the regime where the degree is kept fixed while the number of vertices tends to ∞. The marginal distribution of an individual entry is now well understood, thanks to the work of Bhamidi, van der Hofstad and Hooghiemstra (2010). The purpose of this note is to show that the whole array, suitably recentered, converges in the weak sense to an explicit infinite random array. Our proof consists in analyzing the invasion of the network by several mutually exclusive flows emanating from different sources and propagating simultaneously along the edges.

We prove that the Fourier-Stieltjes transforms of probability measures absolutely continuous with respect to a Rajman measure ? on the circle are uniformly dense in the set of Fourier-Stieltjes transforms of all Rajman probability measures if and only if ? has support the circle.

The purpose of this work is to describe a duality between a fragmentation associated to certain Dirichlet distributions and a natural random coagulation. The dual fragmentation and coalescent chains arising in this setting appear in the description of the genealogy of Yule processes., Comment: 14 pages

Consider a semimartingale of the form $Y_t=Y_0+\int_0^ta_sds+\int_0^t\si_{s-}~dW_s$, where $a$ is a locally bounded predictable process and $\si$ (the ``volatility'') is an adapted right--continuous process with left limits and $W$ is a Brownian motion. We define the realised bipower variation process $V(Y;r,s)^n_t=n^{{r+s\over2}-1}\sum_{i=1}^{[nt]} |Y_{i\over n}-Y_{i-1\over n}|^r|Y_{i+1\over n}-Y_{i\over n}|^s$, where $r$ and $s$ are nonnegative reals with $r+s>0$. We prove that $V(Y;r,s)^n_t$ converges locally uniformly in time, in probability, to a limiting process $V(Y;r,s)_t$ (the ''bipower variation process''). If further $\si$ is a possibly discontinuous semimartingale driven by a Brownian motion which may be correlated with $W$ and by a Poisson random measure, we prove a central limit theorem, in the sense that $\rn~(V(Y;r,s)^n-V(Y;r,s))$ converges in law to a process which is the stochastic integral with respect to some other Brownian motion $W'$, which is independent of the driving terms of $Y$ and $\si$. We also provide a multivariate version of these results.

International audience

International audience; We propose framework for modeling portfolio risk which integrates market risk with liquidation costs which may arise in stress scenarios. Our model provides a systematic method for computing liquidation-adjusted risk measures for a portfolio. Calculation of Liquidation-adjusted VaR (LVaR) for sample portfolios reveals a substantial impact of liquidation costs on portfolio risk for portfolios with large concentrated positions.

International audience; We present a systematic method for computing explicit approximations to martingale representations for a large class of Brownian functionals. The approximations are obtained by obtained by computing a directional derivative of the weak Euler scheme and yield a consistent estimator for the integrand in the martingale representation formula for any square-integrable functional of the solution of an SDE with path-dependent coefficients. Explicit convergence rates are derived for functionals which are Lipschitz-continuous in the supremum norm. Our results require neither the Markov property, nor any differentiability conditions on the functional or the coefficients of the stochastic differential equations involved.

We discuss the Giardin\`a-Kurchan-Peliti population dynamics method for evaluating large deviations of time averaged quantities in Markov processes [Phys. Rev. Lett. \textbf{96}, 120603 (2006)]. This method exhibits systematic errors which can be large in some circumstances, particularly for systems with weak noise, with many degrees of freedom, or close to dynamical phase transitions. We show how these errors can be mitigated by introducing control forces within the algorithm. These forces are determined by an iteration-and-feedback scheme, inspired by multicanonical methods in equilibrium sampling. We demonstrate substantially improved results in a simple model and we discuss potential applications to more complex systems., Comment: 18 pages, 9 figures

In this paper we analyze a wave-based imaging modality called ghost imaging that can produce an image of an object illuminated by a partially coherent source. The image of the object is obtained by correlating the intensities measured by two detectors, one that does not view the object and another one that does view the object. More exactly, a high-resolution detector measures the intensity of a wave field emitted by a partially coherent source which has not interacted with the object to be imaged. A bucket (or single-pixel) detector collects the total (spatially-integrated) intensity of the wave field emitted by the same source that has interacted with the object. The correlation of the intensity measured at the high-resolution detector with the intensity measured by the bucket detector gives an image of the object. In this paper we analyze this imaging modality when the medium through which the waves propagate is random. We discuss the relation with time reversal focusing and with correlation-based imaging using ambient noise sources. We clarify the role of the partial coherence of the source and we study how scattering affects the resolution properties of the ghost imaging function in the paraxial regime: the image resolution is all the better as the source is less coherent, and all the worse as the medium is more scattering.

International audience

We consider large random graphs with prescribed degrees, as generated by the configuration model. In the regime where the empirical degree distribution approaches a limit μ with finite mean, we establish the systematic convergence of a broad class of graph parameters that includes the independence number, the maximum cut size, the logarithm of the Tutte polynomial, and the free energy of the anti-ferromagnetic Ising and Potts models. Contrary to previous works, our results are not a priori limited to the free energy of some prescribed graphical model. They apply more generally to any additive, Lipschitz and concave graph parameter. In addition, the corresponding limits are shown to be Lipschitz and concave in the degree distribution μ. This considerably extends the applicability of the celebrated interpolation method, introduced in the context of spin glasses, and recently related to the challenging question of right-convergence of sparse graphs.

International audience

We consider random dynamics on a uniform random recursive tree with $n$ vertices. Successively, in a uniform random order, each edge is either set on fire with some probability $p_n$ or fireproof with probability $1-p_n$. Fires propagate in the tree and are only stopped by fireproof edges. We first consider the proportion of burnt and fireproof vertices as $n\to\infty$, and prove a phase transition when $p_n$ is of order $\ln n/n$. We then study the connectivity of the fireproof forest, more precisely the existence of a giant component. We finally investigate the sizes of the burnt subtrees., Accepted for publication in Stochastic Processes and their Applications. 24 pages, 4 figures

International audience; This volume contains lecture notes from the courses given by Vlad Bally and Rama Cont at the Barcelona Summer School on Stochastic Analysis (July 2012).The notes of the course by Vlad Bally, co-authored with Lucia Caramellino, develop integration by parts formulas in an abstract setting, extending Malliavin's work on abstract Wiener spaces. The results are applied to prove absolute continuity and regularity results of the density for a broad class of random processes.Rama Cont's notes provide an introduction to the Functional Itô Calculus, a non-anticipative functional calculus that extends the classical Itô calculus to path-dependent functionals of stochastic processes. This calculus leads to a new class of path-dependent partial differential equations, termed Functional Kolmogorov Equations, which arise in the study of martingales and forward-backward stochastic differential equations.This book will appeal to both young and senior researchers in probability and stochastic processes, as well as to practitioners in mathematical finance.

We provide a unified framework for modeling LIBOR rates using general semimartingales as driving processes and generic functional forms to describe the evolution of the dynamics. We derive sufficient conditions for the model to be arbitrage-free which are easily verifiable, and for the LIBOR rates to be true martingales under the respective forward measures. We discuss when the conditions are also necessary and comment on further desirable properties such as those leading to analytical tractability and positivity of rates. This framework allows to consider several popular models in the literature, such as LIBOR market models driven by Brownian motion or jump processes, the Lévy forward price model as well as the affine LIBOR model, under one umbrella. Moreover, we derive structural results about LIBOR models and show, in particular, that only models where the forward price is an exponentially affine function of the driving process preserve their structure under different forward measures.

The Poland-Scheraga model describes the denaturation transition of two complementary - in particular, equally long - strands of DNA, and it has enjoyed a remarkable success both for quantitative modeling purposes and at a more theoretical level. The solvable character of the homogeneous version of the model is one of features to which its success is due. In the bio-physical literature a generalization of the model, allowing different length and non complementarity of the strands, has been considered and the solvable character extends to this substantial generalization. We present a mathematical analysis of the homogeneous generalized Poland-Scheraga model. Our approach is based on the fact that such a model is a homogeneous pinning model based on a bivariate renewal process, much like the basic Poland-Scheraga model is a pinning model based on a univariate, i.e. standard, renewal. We present a complete analysis of the free energy singularities, which include the localization-delocalization critical point and (in general) other critical points that have been only partially captured in the physical literature. We obtain also precise estimates on the path properties of the model., Comment: 45 pages, 7 figures

We extend classical results about the convergence of nearly unstable AR(p) processes to the infinite order case. To do so, we proceed as in recent works about Hawkes processes by using limit theorems for some well chosen geometric sums. We prove that when the coefficients sequence has a light tail, infinite order nearly unstable autoregressive processes behave as Ornstein-Uhlenbeck models. However, in the heavy tail case, we show that fractional diffusions arise as limiting laws for such processes., Comment: 16 pages

This paper deals with the large deviations behavior of a stochastic process called a thinned Lévy process. This process appeared recently as a stochastic-process limit in the context of critical inhomogeneous random graphs (Bhamidi et al. (2012)). The process has a strong negative drift, while we are interested in the rare event of the process being positive at large times. To characterize this rare event, we identify a tilted measure. This presents some challenges inherent to the power-law nature of the thinned Lévy process. General principles prescribe that the tilt should follow from a variational problem, but in the case of the thinned Lévy process this involves a Riemann sum that is hard to control. We choose to approximate the Riemann sum by its limiting integral, derive the first-order correction term, and prove that the tilt that follows from the corresponding approximate variational problem is sufficient to establish the large deviations results.

International audience

We consider a population of N N individuals, of which we observe the number of actions until time t t . For each couple of individuals (i,j) i j , j j may or not influence i i , which we model by i.i.d. Bernoulli(p) p -random variables, for some unknown parameter p∈(0,1] p 0 1 . Each individual acts autonomously at some unknown rate μ>0 μ 0 and acts by mimetism at some rate proportional to the sum of some function φ φ of the ages of the actions of the individuals which influence him. The function φ φ is unknown but assumed, roughly, to be decreasing and with fast decay. The goal of this paper is to estimate p p , which is the main characteristic of the graph of interactions, in the asymptotic N→∞ N ∞ , t→∞ t ∞ . The main issue is that the mean field limit (as N→∞ N ∞ ) of this model is unidentifiable, in that it only depends on the parameters μ μ and pφ p φ . Fortunately, this mean field limit is not valid for large times. We distinguish the subcritical case, where, roughly, the mean number mt m t of actions per individual increases linearly and the supercritical case, where mt m t increases exponentially. Although the nuisance parameter φ φ is non-parametric, we are able, in both cases, to estimate p p without estimating φ φ in a nonparametric way, with a precision of order N−1/2+N1/2m−1t N 1 2 N 1 2 m t 1 , up to some arbitrarily small loss. We explain, using a Gaussian toy model, the reason why this rate of convergence might be (almost) optimal.

International audience

International audience

International audience

We first clarify through classical examples the status of the laws of macroscopic physics as laws of large numbers. We next consider the mirrors model in a finite $d$-dimensional domain and connected to particles reservoirs at fixed chemical potentials. The dynamics is purely deterministic and non-ergodic. We study the macroscopic current of particles in the stationary regime. We show first that when the size of the system goes to infinity, the behaviour of the stationary current of particles is governed by the proportion of orbits crossing the system. This allows to formulate a necessary and sufficient condition on the distribution of the set of orbits that ensures the validity of Fick's law. Using this approach, we show that Fick's law relating the stationary macroscopic current of particles to the concentration difference holds in three dimensions and above. The negative correlations between crossing orbits play a key role in the argument.

Springer Proceedings in Mathematics & Statistics.; International audience

The problem addressed in this paper is the combined detection and localization of a point reflector embedded in a medium by sensor array imaging when the array response matrix is obtained in a noisy environment. We consider additive measurement noise or a clutter noise in the multiple scattering regime. We study a detection test based on reverse-time migration of the array response matrix that is the most powerful for a given false alarm rate and compare it with a test based on the singular values of the response matrix. Moreover, we show that reflector localization should be performed with reverse-time migration rather than any other form of weighted-subspace migration and we give the standard deviation of the localization error.

In this paper we analyze wave propagation in three-dimensional random media. We consider a source with limited spatial and temporal support that generates spherically diverging waves. The waves propagate in a random medium whose fluctuations have small amplitude and correlation radius larger than the typical wavelength but smaller than the propagation distance. In a regime of separation of scales we prove that the wave is modified in two ways by the interaction with the random medium: first, its time profile is affected by a deterministic diffusive and dispersive convolution; second, the wave fronts are affected by random perturbations that can be described in terms of a Gaussian process whose amplitude is of the order of the wavelength and whose correlation radius is of the order of the correlation radius of the medium. Both effects depend on the two-point statistics of the random medium.

We consider an elliptic pseudo-differential equation with a highly oscillating linear potential modeled as a stationary ergodic random field. The random field is a function composed with a centered long-range correlated Gaussian process. In the limiting of vanishing correlation length, the heterogeneous solution converges to a deterministic solution obtained by averaging the random potential. We characterize the deterministic and stochastic correctors. With proper rescaling, the mean-zero stochas- tic corrector converges to a Gaussian random process in probability and weakly in the spatial variables. In addition, for two prototype equations involving the Laplacian and the fractional Laplacian operators, we prove that the limit holds in distribution in some Hilbert spaces. We also determine the size of the deterministic corrector when it is larger than the stochastic corrector. Depending on the correlation structure of the random field and on the singularities of the Green's function, we show that either the deterministic or the random part of the corrector dominates.

The aim of this paper is to develop a mathematical framework for acousto-electromagnetic tomography and to introduce an efficient reconstruction algorithm. In electromagnetic wave imaging, the resolution is limited by the Rayleigh criterion, that is, half the operating wavelength. By mechanically perturbing the medium, we show that it is possible to achieve a significant resolution enhancement. We provide a new inversion formula for the permittivity distribution from cross-correlations between the electromagnetic boundary measurements in the perturbed medium and those in the unperturbed one. We present numerical results to illustrate the resolution and the stability performances of the proposed reconstruction algorithm.

Introduction.-1 Post-crisis fixed-income markets.-1.1 Types of interest rates and market conventions.-1.2 Implications of the crisis.-1.3 The new paradigm: multiple curves at all levels.-1.4 Interest rate derivatives.-2 Short rate models for multiple curves.-2.1 Exponentially affine factor models for the short rate and the Spreads.-2.2 Gaussian, exponential quadratic models.-2.3 Pricing of interest rate derivatives, part A: Pricing of FRAs (linear derivatives).-2.4 Pricing of interest rate derivatives, part B: Pricing of caps (optional derivatives).-2.5 Pricing of interest rate derivatives, part C: Pricing of swaptions (optional derivatives).-2.6 Relationship with models from the literature.-3 Multiple-curve HJM framework.-3.1 Setup and main ideas.-3.2 Absence of arbitrage.-3.3 Volatility structures and the corresponding models.-3.4 Pricing of interest rate derivatives in the HJM framework.-4 LMM multiple-curve extensions.-4.1 Libor market model (LMM) with stochastic basis.-4.2 Affine LIBOR models with multiple curves.-4.3 Cross-currency model analogy.-5 Beyond clean valuation of interest rate derivatives.-5.1 TVA (CVA, DVA, FVA) computations for multiple curve models.-5.2 Direct pricing under collateralization and funding constraints.

We construct a stochastic calculus with respect to the local time process of a symmetric Levy process X without Brownian component. The required assumptions on the Levy process are satisfied by the symmetric stable processes with index in ( 1 , 2 ) . Based on this construction, the explicit decomposition of F ( X t , t ) is obtained for F continuous function admitting a Radon–Nikodym derivative ∂ F ∂ t and satisfying some integrability condition. This Ito formula provides, in particular, the precise expression of the martingale and the continuous additive functional present in Fukushima’s decomposition.

We investigate the functional quantization problem for stochastic processes with respect toLp(IRd, μ)-norms, where μ is a fractal measure namely, μ is self-similar or a homogeneous Cantor measure. The derived functional quantization upper rate bounds are universal depending only on the mean-regularity index of the process and the quantization dimension of μ and as universal rates they are optimal. Furthermore, for arbitrary Borel probability measures μ we establish a (nonconstructive) link between the quantization errors of μ and the functional quantization errors of the process in the spaceLp(IRd, μ).

We propose a continuous-time stochastic model for the dynamics of a limit order book. The model strikes a balance between three desirable features: it can be estimated easily from data, it captures key empirical properties of order book dynamics, and its analytical tractability allows for fast computation of various quantities of interest without resorting to simulation. We describe a simple parameter estimation procedure based on high-frequency observations of the order book and illustrate the results on data from the Tokyo Stock Exchange. Using simple matrix computations and Laplace transform methods, we are able to efficiently compute probabilities of various events, conditional on the state of the order book: an increase in the midprice, execution of an order at the bid before the ask quote moves, and execution of both a buy and a sell order at the best quotes before the price moves. Using high-frequency data, we show that our model can effectively capture the short-term dynamics of a limit order book. We also evaluate the performance of a simple trading strategy based on our results.

International audience; We introduce stochastic models for the transport of heat in systems described by local collisional dynamics. The dynamics consists of tracer particles moving through an array of hot scatterers describing the effect of heat baths at fixed temperatures. Those models have the structure of Markov renewal processes. We study their ergodic properties in details and provide a useful formula for the cumulant generating function of the time integrated energy current. We observe that out of thermal equilibrium, the generating function is not analytic. When the set of temperatures of the scatterers is fixed by the condition that in average no energy is exchanged between the scatterers and the system, different behaviours may arise. When the tracer particles are allowed to travel freely through the whole array of scatterers, the temperature profile is linear. If the particles are locked in between scatterers, the temperature profile becomes nonlinear. In both cases, the thermal conductivity is interpreted as a frequency of collision between tracers and scatterers.

Let F be a separable Banach space, and let (X, Y) be a random pair taking values in F × R. Motivated by a broad range of potential applications, we investigate rates of convergence of the k-nearest neighbor estimate rn (x) of the regression function r(x) = E[Y|X = x], based on n independent copies of the pair (X, Y). Using compact embedding theory, we present explicit and general finite sample bounds on the expected squared difference E[rn(X) - r(X)]2, and particularize our results to classical function spaces such as Sobolev spaces, Besov spaces, and reproducing kernel Hilbert spaces.

International audience; This paper deals with the problem of the multivariate copula density estimation. Using wavelet methods we provide two shrinkage procedures based on thresholding rules for which the knowledge of the regularity of the copula density to be estimated is not necessary. These methods, said to be adaptive, are proved to perform very well when adopting the minimax and the maxiset approaches. Moreover we show that these procedures can be discriminated in the maxiset sense. We produce an estimation algorithm whose qualities are evaluated thanks some simulation. Last, we propose a real life application for financial data.

We consider two simple models for the formation of polymers where at the initial time, each monomer has a certain number of potential links (called arms in the text) that are consumed when aggregations occur. Loosely speaking, this imposes restrictions on the number of aggregations. The dynamics of concentrations are governed by modifications of Smoluchowski's coagulation equations. Applying classical techniques based on generating functions, resolution of quasi-linear PDE's, and Lagrange inversion formula, we obtain explicit solutions to these non-linear systems of ODE's. We also discuss the asymptotic behavior of the solutions and point at some connexions with certain known solutions to Smoluchowski's coagulation equations with additive or multiplicative kernels., To appear in : Annales de l' Institut Henri Poincar\'e Analyse Non-lin\'eaire

We prove a change of variable formula for the 2D fractional Brownian motion of index H bigger of equal to 1/4. For H strictly bigger than 1/4, our formula coincides with that obtained by using the rough paths theory. For H=1/4 (the more interesting case), there is an additional term that is a classical Wiener integral against an independent standard Brownian motion., 16 pages; to appear in Journal of Functional Analysis

Gaussian graphical models are promising tools for analysing genetic networks. However, assessing a network using microarray experiments arises difficult statistical and computational questions. In the present paper, we construct a procedure for testing the neighborhoods of a Gaussian graphical model. Our approach is based on the connection between local Markov property and conditional regression of a Gaussian random variable. Thus, we adapt the testing procedures defined in a preceding paper (N. Verzelen and F.Villers, Goodness-of-fit Tests for high-dimensional Gaussian linear models, Arxiv:math.ST/0711.2119) to this Gaussian graphical modelling framework. Our new tests then inherits appealing theoretical properties. Besides, they apply and are computationally feasible in a high-dimensional setting: the number of observations may be much smaller than the number of nodes. A large part of this study is deserved to illustrate and discuss the application of our procedures to simulated data and to biological data.

In this article, we study the family of probability measures (indexed by t ∈ R + ∗ ), obtained by penalization of the Brownian motion with a given functional of its local times at time t . We prove that this family tends to a limit measure when t goes to infinity if the functional satisfies some conditions of domination, and we check these conditions in several particular cases.

We show that every ergodic, finite entropy transformation which satisfies the weak Pinsker property possesses a finite generator whose two-sided tail field is exactly the Pinsker algebra. This is proved by exhibiting a generator endowed with a block structure quite analogous to the one appearing in the construction of the Ornstein–Shields examples of non Bernoulli K-automorphisms. We also show that, given two transformations T1 and T2 in the previous class (i.e. satisfying the weak Pinsker property), and a Bernoulli shift B, if T1×B is isomorphic to T2×B, then T1 is isomorphic to T2. That is, one can ‘factor out’ Bernoulli shifts.