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Nonparametric adaptive estimation for pure jump Lévy processes

F. Comte, V. Genon-Catalot (2010)

Annales de l'I.H.P. Probabilités et statistiques

This paper is concerned with nonparametric estimation of the Lévy density of a pure jump Lévy process. The sample path is observed at n discrete instants with fixed sampling interval. We construct a collection of estimators obtained by deconvolution methods and deduced from appropriate estimators of the characteristic function and its first derivative. We obtain a bound for the -risk, under general assumptions on the model. Then we propose a penalty function that allows to build an adaptive estimator....

Non-parametric approximation of non-anticipativity constraints in scenario-based multistage stochastic programming

Jean-Sébastien Roy, Arnaud Lenoir (2008)

Kybernetika

We propose two methods to solve multistage stochastic programs when only a (large) finite set of scenarios is available. The usual scenario tree construction to represent non-anticipativity constraints is replaced by alternative discretization schemes coming from non-parametric estimation ideas. In the first method, a penalty term is added to the objective so as to enforce the closeness between decision variables and the Nadaraya–Watson estimation of their conditional expectation. A numerical application...

Nonparametric estimation of the derivatives of the stationary density for stationary processes

Emeline Schmisser (2013)

ESAIM: Probability and Statistics

In this article, our aim is to estimate the successive derivatives of the stationary density f of a strictly stationary and β-mixing process (Xt)t≥0. This process is observed at discrete times t = 0,Δ,...,nΔ. The sampling interval Δ can be fixed or small. We use a penalized least-square approach to compute adaptive estimators. If the derivative f(j)belongs to the Besov space B 2 , ∞ α B 2 , ∞ α , then our estimator converges at rate (nΔ)−α/(2α+2j+1). Then we consider a diffusion with known diffusion coefficient....

Nonparametric inference for discretely sampled Lévy processes

Shota Gugushvili (2012)

Annales de l'I.H.P. Probabilités et statistiques

Given a sample from a discretely observed Lévy process X = (Xt)t≥0 of the finite jump activity, the problem of nonparametric estimation of the Lévy density ρ corresponding to the process X is studied. An estimator of ρ is proposed that is based on a suitable inversion of the Lévy–Khintchine formula and a plug-in device. The main results of the paper deal with upper risk bounds for estimation of ρ over suitable classes of Lévy triplets. The corresponding lower bounds are also discussed.

Non-stationary departure process in a batch-arrival queue with finite buffer capacity and threshold-type control mechanism

Wojciech M. Kempa, Dariusz Kurzyk (2022)

Kybernetika

Non-stationary behavior of departure process in a finite-buffer M X / G / 1 / K -type queueing model with batch arrivals, in which a threshold-type waking up N -policy is implemented, is studied. According to this policy, after each idle time a new busy period is being started with the N th message occurrence, where the threshold value N is fixed. Using the analytical approach based on the idea of an embedded Markov chain, integral equations, continuous total probability law, renewal theory and linear algebra, a...

Non-symmetric hitting distributions on the hyperbolic half-plane and subordinated perpetuities.

Paolo Baldi, Enrico Casadio Tarabusi, Alessandro Figà-Talamanca, Marc Yor (2001)

Revista Matemática Iberoamericana

We study the law of functionals whose prototype is ∫0+∞ eBs(ν) dWs(μ),where B(ν) and W(μ) are independent Brownian motions with drift. These functionals appear naturally in risk theory as well as in the study of in variant diffusions on the hyperbolic half-plane. Emphasis is put on the fact that the results are obtained in two independent, very different fashions (invariant diffusions on the hyperbolic half-plane and Bessel processes).

Normal martingales and polynomial families

H. Hammouch (2004)

Annales Polonici Mathematici

Wiener and compensated Poisson processes, as normal martingales, are associated to classical sequences of polynomials, namely Hermite polynomials for the first one and Charlier polynomials for the second. The problem studied in this paper is to find if there exist other normal martingales which are associated to classical sequences of polynomials. Privault, Solé and Vives [5] solved this problem via the quantum Kabanov formula under some assumptions on the normal martingales considered. We solve...

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