Stochastic analysis for Poisson processes

  title={Stochastic analysis for Poisson processes},
  author={Gunter Last},
  • G. Last
  • Published 17 May 2014
  • Mathematics
This survey is a preliminary version of a chapter of the forthcoming book [21]. The paper develops some basic theory for the stochastic analysis of Poisson process on a general σ-finite measure space. After giving some fundamental definitions and properties (as the multivariate Mecke equation) the paper presents the Fock space representation of square-integrable functions of a Poisson process in terms of iterated difference operators. This is followed by the introduction of multivariate… 

Stable limit theorems on the Poisson space

  • Ronan Herry
  • Mathematics
    Electronic Journal of Probability
  • 2020
We prove limit theorems for functionals of a Poisson point process using the Malliavin calculus on the Poisson space. The target distribution is either a conditional Gaussian vector or a conditional

The fourth moment theorem on the Poisson space

We prove an exact fourth moment bound for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result -- that has been elusive for

Poisson Malliavin calculus in Hilbert space with an application to SPDE

In this paper we introduce a Hilbert space-valued Malliavin calculus for Poisson random measures. It is solely based on elementary principles from the theory of point processes and basic moment

A four moments theorem for Gamma limits on a Poisson chaos

This paper deals with sequences of random variables belonging to a fixed chaos of order $q$ generated by a Poisson random measure on a Polish space. The problem is investigated whether convergence of

Hyperbolic Anderson model with L\'evy white noise: spatial ergodicity and fluctuation

In this paper, we study one-dimensional hyperbolic Anderson models (HAM) driven by space-time L\'evy white noise in a finite-variance setting. Motivated by recent active research on limit theorems

Normal Approximation of Poisson Functionals in Kolmogorov Distance

  • Matthias Schulte
  • Mathematics, Computer Science
    Journal of Theoretical Probability
  • 2014
This paper shows that convergence in the Wasserstein distance of a Poisson functional and a Gaussian random variable has the same rate for both distances for a large class of Poisson functionals, namely so-called U-statistics ofPoisson point processes.

Normal Approximation of Poisson Functionals in Kolmogorov Distance

This paper shows that convergence in the Wasserstein distance of a Poisson functional and a Gaussian random variable has the same rate for both distances for a large class of Poisson functionals, namely so-called U-statistics ofPoisson point processes.

Cluster Expansions for GIBBS Point Processes

  • S. Jansen
  • Mathematics
    Advances in Applied Probability
  • 2019
Abstract We provide a sufficient condition for the uniqueness in distribution of Gibbs point processes with non-negative pairwise interaction, together with convergent expansions of the log-Laplace

Poisson fluctuations for edge counts in high-dimensional random geometric graphs

We prove a Poisson limit theorem in the total variation distance of functionals of a general Poisson point process using the Malliavin-Stein method. Our estimates only involve first and second order

Malliavin calculus for marked binomial processes: portfolio optimisation in the trinomial model and compound Poisson approximation

In this paper we develop a stochastic analysis for marked binomial processes, that can be viewed as the discrete analogues of marked Poisson processes. The starting point is the statement of a

Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with

Orthogonal functionals of the Poisson process

  • H. Ogura
  • Mathematics
    IEEE Trans. Inf. Theory
  • 1972
It is shown that any nonlinear functional of the Poisson process with finite variance can be developed in terms of these orthogonal functionals, corresponding to the Cameron-Martin theorem in the case of the Brownian-motion process.

Gamma limits and U-statistics on the Poisson space

Using Stein's method and the Malliavin calculus of variations, we derive explicit estimates for the Gamma approximation of functionals of a Poisson measure. In particular, conditions are presented

Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization

We prove a new class of inequalities, yielding bounds for the normal approximation in the Wasserstein and the Kolmogorov distance of functionals of a general Poisson process (Poisson random measure).

Mini-Workshop : Stochastic Analysis for Poisson Point Processes : Malliavin Calculus , Wiener-Ito Chaos Expansions and Stochastic Geometry

Malliavin calculus plays an important role in the stochastic analysis for Poisson point processes. This technique is tightly connected with chaotic expansions, that were introduced in the first half

On the existence of smooth densities for jump processes

SummaryWe consider a Lévy process Xt and the solution Yt of a stochastic differential equation driven by Xt; we suppose that Xt has infinitely many small jumps, but its Lévy measure may be very

On multiple Poisson stochastic integrals and associated Markov semigroups

with respect to the centered Poisson random measure q(dx), E[q(dx)] = 0, E[(q(dx))] = m(dx), are discussed, where (X, m) is a measurable space. A ”diagram formula” for evaluation of products of

On homogeneous chaos

  • N. CutlandSiu-Ah Ng
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1991
Abstract This paper discusses the Wiener–Itô chaos decomposition of an L2 function φ over Wiener space, and is concerned in particular with the identification of the integrands ƒn in the chaos

Concentration and deviation inequalities in infinite dimensions via covariance representations

Concentration and deviation inequalities are obtained for functionals on Wiener space, Poisson space or more generally for normal martingales and binomial processes. The method used here is based on