Christian Houdré

Risk bounds for the non-parametric estimation of Lévy processes

Estimation methods for the Levy density of a Levy process are developed under mild qualitative assumptions. A classical model selection ap- proach made up of two steps is studied. The first step consists in the selection of a good estimator, from an approximating (finite-dimensional) linear model S for the true Levy density. The second is a data-driven selection of a linear model S, among a given collection {Sm}m∈M, that approximately realizes the best trade-off between the error of estimation within S and the error incurred when approximating the true Levy density by the linear model S. Using recent concentration inequalities for functionals of Poisson integrals, a bound for the risk of estimation is obtained. As a byproduct, oracle inequalities and long- run asymptotics for spline estimators are derived. Even though the resulting underlying statistics are based on continuous time observations of the process, approximations based on high-frequency discrete-data can be easily devised.

Exponential Inequalities, with Constants, for U-statistics of Order Two

A martingale proof of a sharp exponential inequality (with constants) is given for U-statistics of order two as well as for double integrals of Poisson processes.

On Gaussian and Bernoulli covariance representations

We discuss several applications, to large deviations for smooth functions of Gaussian random vectors, of a covariance representation in Gauss space. The existence of this type of representation characterizes Gaussian measures. New representations for Bernoulli measures are also derived, recovering some known inequalities.

Interpolation, correlation identities, and inequalities for infinitely divisible variables

We present an interpolation formula for the expectation of functions of infinitely divisible (i.d.) variables. This is then applied to study the association problem for i.d. vectors and to present new covariance expansions and correlation inequalities.

On layered stable processes

^School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA. E-mail: houdre@math.gatech.edu 2Financial Engineering, Fixed Income Department, Daiwa Securities SMBC Co. Ltd., 1-14-5 Eitai, Koto-ku, Tokyo 135-0034, Japan. E-mail: reiichiro.kawai@daiwasmbc.co.jp. The views expressed in this paper are those of the author and do not reflect those of Daiwa Securities SMBC Co.Ltd.

On the Concentration of Measure Phenomenon for Stable and Related Random Vectors

where m(f(X)) is a median of f(X) and where Φ is the (one-dimensional) standard normal distribution function. The inequality (1) has seen many extensions and to date, most of the conditions under which these developments hold require the existence of finite exponential moments for the underlying vector X . It is thus natural to explore the robustness of this “concentration phenomenon” and to study the corresponding results for stable vectors. It is the purpose of these notes to initiate this study and to present a few concentration results for stable and related vectors, freeing us from the exponential moment requirement. Our main result will imply that if X is an α-stable random vector in Rd, then for all x> 0,

Small-time expansions of the distributions, densities, and option prices of stochastic volatility models with Lévy jumps

We consider a stochastic volatility model with Levy jumps for a log-return process Z=(Zt)t≥0 of the form Z=U+X, where U=(Ut)t≥0 is a classical stochastic volatility process and X=(Xt)t≥0 is an independent Levy process with absolutely continuous Levy measure ν. Small-time expansions, of arbitrary polynomial order, in time-t, are obtained for the tails P(Zt≥z), z>0, and for the call-option prices E(ez+Zt−1)+, z≠0, assuming smoothness conditions on the density of ν away from the origin and a small-time large deviation principle on U. Our approach allows for a unified treatment of general payoff functions of the form φ(x)1x≥z for smooth functions φ and z>0. As a consequence of our tail expansions, the polynomial expansions in t of the transition densities ft are also obtained under mild conditions.

On the Longest Increasing Subsequence for Finite and Countable Alphabets

Let X1; X2; : : : ; Xn; : : : be a sequence of iid random variables with values in a nite alphabet f1; : : : ; mg. Let LIn be the length of the longest increasing subsequence of X1; X2; : : : ; Xn: We express the limiting distribution of LIn as functionals of m and (m 1)-dimensional Brownian motions. These expressions are then related to similar functionals appearing in queueing theory, allowing us to further establish asymptotic behaviors as m grows. The nite alphabet results are then used to treat the countable (innite) alphabet.

Mixed and Isoperimetric Estimates on the Log-Sobolev Constants of Graphs and Markov Chains

Two types of lower bounds are obtained on the log-Sobolev constants of graphs and Markov chains. The first is a mixture of spectral gap and logarithmic isoperimetric constant, the second involves the Gaussian isoperimetric constant. The sharpness of both types of bounds is tested on some examples. Product generalizations of some of these results are also briefly given.

Some Applications of Covariance Identities and Inequalities to Functions of Multivariate Normal Variables

Abstract We apply general covariance identities and inequalities to some functions of multivariate normal variables. We recover, in particular, a recent covariance identity due to Siegel and provide simple estimates on the variance of order statistics. We also present some computations.