Simulations of some Doubly Stochastic Poisson Point Processes

International audienceComputer simulations of point processes are important either to verify the results of certain theoretical calculations that can be very awkward at times, or to obtain practical results when these calculations become almost impossible. One of the most common methods for the simulation of nonstationary Poisson processes is random thinning. Its extension when the intensity becomes random (doubly stochastic Poisson processes) depends on the structure of this intensity. If the random density takes only discrete values, which is a common situation in many physical problems where quantum mechanics introduces discrete states, it is shown that the thinning method can be applied without error. We study in particular the case of binary density and we present the kind of theoretical calculations that then become possible. The results of various experiments realized with data obtained by simulation show fairly good agreement with the theoretical calculations

Fractional Poisson Fields and Martingales

We present new properties for the Fractional Poisson process and the Fractional Poisson field on the plane. A martingale characterization for Fractional Poisson processes is given. We extend this result to Fractional Poisson fields, obtaining some other characterizations. The fractional differential equations are studied. We consider a more general Mixed-Fractional Poisson process and show that this process is the stochastic solution of a system of fractional differential-difference equations. Finally, we give some simulations of the Fractional Poisson field on the plane

Testing the existence of clustering in the extreme values

This paper introduces an estimator for the extremal index as the ratio of the number of elements of two point processes defined by threshold sequences un, vn and a partition of the sequence in different blocks of the same size. The first point process is defined by the sequence of the block maxima that exceed un. This paper introduces a thinning of this point process, defined by a threshold vn with vn > un, and with the appealing property that under some mild conditions the ratio of the number of elements of both point processes is a consistent estimator of the extremal index. The method supports a hypothesis test for the extremal index, and hence for testing the existence of clustering in the extreme values. Other advantages are that it allows some freedom to choose un, and it is not very sensitive to the choice of the partition. Finally, the stylized facts found in financial returns (clustering, skewness, heavy tails) are tested via the extremal index, in this case for the DaX return

Stochastic foundations of undulatory transport phenomena: Generalized Poisson-Kac processes - Part I Basic theory

This article introduces the notion of Generalized Poisson-Kac (GPK) processes which generalize the class of "telegrapher's noise dynamics" introduced by Marc Kac in 1974, usingPoissonian stochastic perturbations. In GPK processes the stochastic perturbation acts as a switching amongst a set of stochastic velocity vectors controlled by a Markov-chain dynamics. GPK processes possess trajectory regularity (almost everywhere) and asymptotic Kac limit, namely the convergence towards Brownian motion (and to stochastic dynamics driven by Wiener perturbations), which characterizes also the long-term/long-distance properties of these processes. In this article we introduce the structural properties of GPK processes, leaving all the physical implications to part II and part III

Nonparametric inference of doubly stochastic Poisson process data via the kernel method

Doubly stochastic Poisson processes, also known as the Cox processes, frequently occur in various scientific fields. In this article, motivated primarily by analyzing Cox process data in biophysics, we propose a nonparametric kernel-based inference method. We conduct a detailed study, including an asymptotic analysis, of the proposed method, and provide guidelines for its practical use, introducing a fast and stable regression method for bandwidth selection. We apply our method to real photon arrival data from recent single-molecule biophysical experiments, investigating proteins' conformational dynamics. Our result shows that conformational fluctuation is widely present in protein systems, and that the fluctuation covers a broad range of time scales, highlighting the dynamic and complex nature of proteins' structure.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS352 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Numerical approximation of doubly reflected BSDEs with jumps and RCLL obstacles

We study a discrete time approximation scheme for the solution of a doubly reflected Backward Stochastic Differential Equation (DBBSDE in short) with jumps, driven by a Brownian motion and an independent compensated Poisson process. Moreover, we suppose that the obstacles are right continuous and left limited (RCLL) processes with predictable and totally inaccessible jumps and satisfy Mokobodski's condition. Our main contribution consists in the construction of an implementable numerical sheme, based on two random binomial trees and the penalization method, which is shown to converge to the solution of the DBBSDE. Finally, we illustrate the theoretical results with some numerical examples in the case of general jumps