# Stochastic analysis for Poisson processes

@inproceedings{Last2014StochasticAF, title={Stochastic analysis for Poisson processes}, author={Gunter Last}, year={2014} }

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…

## 44 Citations

### Stable limit theorems on the Poisson space

- MathematicsElectronic 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

- MathematicsThe Annals of Probability
- 2018

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

- Mathematics
- 2017

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

- Mathematics
- 2015

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

- Mathematics
- 2023

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

- Mathematics, Computer ScienceJournal 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

- Mathematics, Computer Science
- 2012

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

- MathematicsAdvances 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

- Mathematics
- 2019

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

- Mathematics
- 2021

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…

## 29 References

### Moments and Central Limit Theorems for Some Multivariate Poisson Functionals

- MathematicsAdvances in Applied Probability
- 2014

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

- MathematicsIEEE 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

- Mathematics
- 2013

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

- Mathematics
- 2014

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

- Mathematics
- 2013

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…

### Martingale representation for Poisson processes with applications to minimal variance hedging

- Mathematics
- 2010

### On the existence of smooth densities for jump processes

- Mathematics
- 1996

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

- Mathematics
- 1984

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

- MathematicsMathematical 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

- Mathematics
- 2002

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…