We establish some error estimates for the approximation of anoptimal stopping problem along the paths of the Black–Scholesmodel. This approximation is based on a tree method. Moreover, wegive a global approximation result for the related obstacleproblem.

Let F be a distribution function (d.f) in the domain of attraction of an extreme value distribution ${ H_{\gamma} }$; it is well-known that Fu(x), where Fu is the d.f of the excesses over u, converges, when u tends to s+(F), the end-point of F, to $G_{\gamma}(\frac{x}{\sigma(u)})$, where $G_{\gamma}$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma >-1$, a function Λ which verifies $\lim_{u \rightarrow s_+(F)} \Lambda (u) =\gamma$ and is such that $\Delta(u)= \sup_{x \in[0,s_+(F)-u[} |\bar{F}_u(x) - \bar{G}_{\Lambda(u)} (x/ \sigma(u))| $converges to 0 faster than $d(u)=\sup_{x \in[0,s_+(F)-u[} |\bar{F}_u(x) - \bar{G}_{\gamma}(x/ \sigma(u))|$.

In this paper, we are interested in the asymptotical behaviorof the error between the solution of a differential equation perturbed by a flow (or by a transformation) and the solution of the associated averaged differential equation.The main part of this redaction is devoted to the ascertainment of results of convergence in distribution analogous to those obtained in [10] and [11]. As in [11], we shall use a representation by a suspension flow over a dynamical system. Here, we make an assumption of multiple decorrelation in terms of this dynamical system. We show how this property can be verified for ergodic algebraic toral automorphisms and point out the fact that, fortwo-dimensional dispersive billiards, it is a consequence of the method developed in [18]. Moreover, the singular case of a degenerated limit distribution is also considered.

We discuss various properties of Probabilistic Cellular Automata, suchas the structure of the set of stationary measures and multiplicity ofstationary measures (or phase transition) for reversible models.

In this paper two characterizations of the Pólyadistribution are obtained when its contagion parameter isnegative. One of them is based on mixtures and the other one isobtained by characterizing a subfamily of the discrete Pearsonsystem.

We investigate the steady state behavior of an M/G/1 queue with modified Bernoulli schedule server vacations. Batches of variable size arrive at the system according to a compound Poisson process. However, all arriving batches are not allowed into the system. The restriction policy differs when the server is available in the system and when he is on vacation. We obtain in closed form, the steady state probability generating functions for the number of customers in the queue for various states of the server, the average number of customers as well as their average waiting time in the queue and the system. Many special cases of interest including complete admissibility, partial admissibility and no server vacations have been discusssed. Some known results are derived as particular cases of our model.

We consider the problem of estimating an unknown regression function when the design is random with values in $\mathbb{R}^k$. Our estimation procedure is based on model selection and does not rely on any prior information on the target function. We start with a collection of linear functional spaces and build, on a data selected space among this collection, the least-squares estimator. We study the performance of an estimator which is obtained by modifying this least-squares estimator on a set of small probability. For the so-defined estimator, we establish nonasymptotic risk bounds that can be related to oracle inequalities. As a consequence of these, we show that our estimator possesses adaptive properties in the minimax sense over large families of Besov balls Bα,l,∞(R) with R>0, l ≥ 1 and α > α1 where α1 is a positive number satisfying 1/l - 1/2 ≤ α1 < 1/l. We also study the particular case where the regression function is additive and then obtain an additive estimator which converges at the same rate as it does when k=1.


Consider an infinite dimensionaldiffusion process process on TZd, whereT is the circle, defined by the action of its generatorL on C2(TZd) local functions as$Lf(\eta)=\sum_{i\in{\bf Z}^d}\left(\frac{1}{2}a_i\frac{\partial^2 f}{\partial \eta_i^2}+b_i\frac{\partial f}{\partial \eta_i}\right)$. Assume that the coefficients, ai and bi are smooth,bounded, finite range with uniformly bounded second orderpartial derivatives, that ai is only a function of $\eta_i$and that $\inf_{i,\eta}a_i(\eta)>0$.Suppose ν is an invariant product measure. Then,if ν is the Lebesgue measure or if d=1,2, it is theunique invariant measure. Furthermore, if ν istranslation invariant, then it is the unique invariant, translationinvariant measure. Now, consider an infinite particlespin system, with state space {0,1}Zd, definedby the action of its generator on local functions f by$Lf(\eta)=\sum_{x\in{\bf Z}^d}c(x,\eta)(f(\eta^x)-f(\eta))$,where $\eta^x$ is the configuration obtained from ηaltering only the coordinate at site x. Assume that $c(x,\eta)$are of finite range, bounded and that $\inf_{x,\eta}c(x,\eta)>0$.Then, if ν is an invariant product measure for this process,ν is unique when d=1,2. Furthermore, if ν is translationinvariant, it is the unique invariant, translation invariantmeasure. The proofs of these results show how elementarymethods can give interesting information for general processes.

In this paper we define and study self-similar rankedfragmentations. We first show that any ranked fragmentation is theimage of some partition-valued fragmentation, and that there is infact a one-to-one correspondence between the laws of these twotypes of fragmentations. We then give an explicit construction ofhomogeneous ranked fragmentations in terms of Poisson pointprocesses. Finally we use this construction and classical resultson records of Poisson point processes to study the small-timebehavior of a ranked fragmentation.

We study the tails of the distribution of the maximum of a stationaryGaussian process on a bounded interval of the real line. Under regularityconditions including the existence of the spectral moment of order 8, we give an additional term for this asymptotics. This widens the application of an expansion given originally by Piterbarg [CITE] for a sufficiently small interval.

This paper is devoted to the study of some asymptotic properties of aM-estimator in a framework of detection of abrupt changes inrandom field's distribution. This class of problems includes e.g.recovery of sets. It involves various techniques, including M-estimation method, concentrationinequalities, maximal inequalities for dependent random variables and ϕ-mixing. Penalization of the criterion function when the size of thetrue model is unknown is performed. All the results apply under mild, discussedassumptions. Simple examples are provided.

In this paper, we study the problem of non parametric estimationof the stationary marginal density f of an α or aβ-mixing process, observed either in continuous time or indiscrete time. We present an unified framework allowing to dealwith many different cases. We consider a collection of finitedimensional linear regular spaces. We estimate f using aprojection estimator built on a data driven selected linear spaceamong the collection. This data driven choice is performed via theminimization of a penalized contrast. We state non asymptotic riskbounds, regarding to the integrated quadratic risk, for ourestimators, in both cases of mixing. We show that they areadaptive in the minimax sense over a large class of Besov balls.In discrete time, we also provide a result for model selectionamong an exponentially large collection of models (non regularcase).

In this paper, we make use of the information measure introducedby Mokkadem (1997) for building a goodness-of-fit test forlong-range dependent processes.Our test statistic is performed in the frequency domain and writes asa non linear functional of the normalized periodogram. We establishthe asymptotic distribution of our statistic under the nullhypothesis. Under specific alternative hypotheses, we prove that the powerconverges to one. The performance of our test procedure isillustrated from different simulated series. In particular,we compare its size and its power with test of Chenand Deo.

In Francq and Zakoïan [4], we derived stationarity conditions forARMA(p,q) models subject to Markov switching. In this paper, weshow that, under appropriate moment conditions, the powers of thestationary solutions admit weak ARMA representations, which we areable to characterize in terms of p,q, the coefficients of themodel in each regime, and the transition probabilities of theMarkov chain. These representations are potentially useful forstatistical applications.

This paper deals with semiparametric convolution models, where thenoise sequence has a Gaussian centered distribution, with unknownvariance. Non-parametric convolution models are concerned with the case of anentirely known distribution for the noise sequence, and they havebeen widely studied in the past decade. The main property of thosemodels is the following one: the more regular the distribution of thenoise is, the worst the rate of convergence for the estimation of thesignal's density g is [5]. Nevertheless, regularity assumptionson the signal density g improve those rates of convergence [15].In this paper, we show that whenthe noise (assumed tobe Gaussian centered) has a varianceσ2 that is unknown (actually, it is always the case inpractical applications), the rates of convergence for the estimation ofg are seriously deteriorated, whatever its regularity is supposed to be.More precisely, the minimax risk for the pointwise estimation of g over aclass of regular densities is lower bounded by a constant over log n. We construct two estimators of σ2, andmore particularly, an estimator which is consistent as soon as the signal hasa finite first order moment.We also mention as a consequence the deterioration of therate of convergence in the estimation of the parameters in the nonlinearerrors-in-variables model.

We study the asymptotic behavior of the empirical process when theunderlying data are Gaussian and exhibit seasonallong-memory. We prove that the limiting process can be quitedifferent from the limit obtained in the case of regularlong-memory. However, in both cases, the limiting process isdegenerated. We apply our results to von–Mises functionals andU-Statistics.

The EGARCH model of Nelson [29] is one of the most successfulARCH models which may exhibit characteristic asymmetries offinancial time series, as well as long memory. The paper studiesthe covariance structure and dependence properties of the EGARCHand some related stochastic volatility models. We show that thelarge time behavior of the covariance of powers of the (observed)ARCH process is determined by the behavior of the covariance ofthe (linear) log-volatility process; in particular, a hyperbolicdecay of the later covariance implies a similar hyperbolic decayof the former covariances. We show, in this case, that normalizedpartial sums of powers of the observed process tend to fractionalBrownian motion. The paper also obtains a (functional) CLT for thecorresponding partial sums' processes of the EGARCH model withshort and moderate memory. These results are applied to studyasymptotic behavior of tests for long memory using the R/Sstatistic.