Motivated by second order asymptotic results, we characterize the convergence in law of double integrals, with respect to Poisson random measures, toward a standard Gaussian distribution. Our conditions are expressed in terms of contractions of the kernels. To prove our main results, we use the theory of stable convergence of generalized stochastic integrals developed by Peccati and Taqqu. One of the advantages of our approach is that the conditions are expressed directly in terms of the kernel appearing in the multiple integral and do not make any explicit use of asymptotic dependence properties such as mixing. We illustrate our techniques by an application involving linear and quadratic functionals of generalized Ornstein--Uhlenbeck processes, as well as examples concerning random hazard rates.
Tweedie exponential dispersion family constitutes a fairly rich sub-class of the celebrated exponential family. In particular, a member, compound Poisson gamma (CP-g) model has seen extensive use over the past decade for modeling mixed response featuring exact zeros with a continuous response from a gamma distribution. This paper proposes a framework to perform residual analysis on CP-g double generalized linear models for spatial uncertainty quantification. Approximations are introduced to proposed framework making the procedure scalable, without compromise in accuracy of estimation and model…
We propose Mecke-Palm formulas for multiple integrals with respect to a Poisson random measure interlaced with its intensity measure. We apply such formulas to multiple mixed Lévy systems of Lévy processes and obtain moment formulas for products of interlaced multiple Poisson integrals.
Let $\tilde{N}\_{t}$ be a standard compensated Poisson process on $[0,1]$. We prove a new characterization of anticipating integrals of the Skorohod type with respect to $\tilde{N}$, and use it to obtain several counterparts to well established properties of semimartingale stochastic integrals. In particular we show that, if the integrand is sufficiently regular, anticipating Skorohod integral processes with respect to $\tilde{N}$ admit a pointwise representation as usual Itô integrals in an independently enlarged filtration. We apply such a result to: (i) characterize Skorohod integral proces…
We compute the moment of order n of the Poisson stochastic integral of a random process u over a metric space X as a sum that runs over all partitions of {1,...,n} and involves the addition of points to Poisson configurations. This formula recovers known results in case u is a deterministic function on X.
Consider a surplus process which both of collected premium and payed claim size are two independent compound Poisson processes. This article derives two approximated formulas for the ruin probability of such surplus process, say double stochastic compound poisson process. More precisely, it provides two mixture exponential approximations for ruin probability of such double stochastic compound poisson process. Applications to long_term Bonus_Malus systems and a heavy-tiled claim size distribution have been given. Improvement of our findings compared to the Cramer- Lundberg upper bound has been …
In an M-type 2 Banach space, firstly we explore some properties of the set-valued stochastic integral associated with the stationary Poisson point process. By using the Hahn decomposition theorem and bounded linear functional, we obtain the main result: the integral of a set-valued stochastic process with respect to the compensated Poisson measure is a set-valued submartingale but not a martingale unless the integrand degenerates into a single-valued process. Secondly we study the strong solution to the set-valued stochastic integral equation, which includes a set-valued drift, a single-valued…
We want to compute the cumulative distribution function of a one-dimensional Poisson stochastic integral $I(\krnl) = \displaystyle \int_0^T \krnl(s) N(ds)$, where $N$ is a Poisson random measure with control measure $n$ and $\krnl$ is a suitable kernel function. We do so by combining a Kolmogorov-Feller equation with a finite-difference scheme. We provide the rate of convergence of our numerical scheme and illustrate our method on a number of examples. The software used to implement the procedure is available on demand and we demonstrate its use in the paper.
In this paper we investigate classical solution of a semi-linear system of backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. By proving an Itô-Wentzell formula for jump diffusions as well as an abstract result of stochastic evolution equations, we obtain the stochastic integral partial differential equation for the inverse of the stochastic flow generated by a stochastic differential equation driven by a Brownian motion and a Poisson point process. By composing the random field generated by the solution of a backward stochastic…
In this paper we study the properties of the Poisson random measure and the Poisson integral associated with a G-Levy process. We prove that a Poisson integral is a G-Levy process and give the conditions which ensure that a Poisson integral belongs to a good space of random variables. In particular, we study the relation between the quasi- continuity of an integrand and the quasi-continuity of the integral. Lastly, we apply the results to establish the pathwise decomposition of a G-Levy process into a generalized G-Brownian motion and a pure-jump G-Levy process and prove that both processes be…
We give a extensive account of a recent new way of applying the Dirichlet form theory to random Poisson measures. The main application is to obtain existence of density for thelaws of random functionals of Lévy processes or solutions of stochastic differential equations with jumps. As in the Wiener case the Dirichlet form approach weakens significantly theregularity assumptions. The main novelty is an explicit formula for the gradient or for the "carré du champ' on the Poisson space called the lent particle formula because based on adding a new particle to the system, computing the derivative …
We prove new upper and lower bounds for Banach space-valued stochastic integrals with respect to a compensated Poisson random measure. Our estimates apply to Banach spaces with non-trivial martingale (co)type and extend various results in the literature. We also develop a Malliavin framework to interpret Poisson stochastic integrals as vector-valued Skorohod integrals, and prove a Clark-Ocone representation formula.
In this paper we consider the Riemann--Liouville fractional integral $\mathcal{N}^{α,ν}(t)= \frac{1}{Γ(α)} \int_0^t (t-s)^{α-1}N^ν(s) \, \mathrm ds $, where $N^ν(t)$, $t \ge 0$, is a fractional Poisson process of order $ν\in (0,1]$, and $α> 0$. We give the explicit bivariate distribution $\Pr \{N^ν(s)=k, N^ν(t)=r \}$, for $t \ge s$, $r \ge k$, the mean $\mathbb{E}\, \mathcal{N}^{α,ν}(t)$ and the variance $\mathbb{V}\text{ar}\, \mathcal{N}^{α,ν}(t)$. We study the process $\mathcal{N}^{α,1}(t)$ for which we are able to produce explicit results for the conditional and absolute variances and means…
We prove It{ô}'s formula for the flow of measures associated with a jump process defined by a drift, an integral with respect to a Poisson random measure and with respect to the associated compensated Poisson random measure. We work in $\mathcal{P}_β(\mathbb{R}^d)$, the space of probability measures on $\mathbb{R}^d$ having a finite moment of order $β\in (0, 2]$. As an application, we exhibit the backward Kolmogorov partial differential equation stated on $[0,T] \times \mathcal{P}_β(\mathbb{R}^d)$ associated with a McKean-Vlasov stochastic differential equation driven by a Poisson random measu…
We obtain Stein approximation bounds for stochastic integrals with respect to a Poisson random measure over ${\Bbb R}^d$, $d\geq 2$. This approach relies on third cumulant Edgeworth-type expansions based on derivation operators defined by the Malliavin calculus for Poisson random measures. The use of third cumulants can exhibit faster convergence rates than the standard Berry-Esseen rate for some sequences of Poisson stochastic integrals.
Motivated by the study of existence, uniqueness and regularity of solutions to stochastic partial differential equations driven by jump noise, we prove Itô isomorphisms for $L^p$-valued stochastic integrals with respect to a compensated Poisson random measure. The principal ingredients for the proof are novel Rosenthal type inequalities for independent random variables taking values in a (noncommutative) $L^p$-space, which may be of independent interest. As a by-product of our proof, we observe some moment estimates for the operator norm of a sum of independent random matrices.
It is interesting and challenging to study double-ended queues with First-Come-First-Match discipline under customers' impatient behavior and non-Poisson inputs. The system stability can be guaranteed by the customers' impatient behavior, while the existence of impatient customers makes analysis of such double-ended queues more difficult or even impossible to find an explicitly analytic solution, thus it becomes more and more important to develop effective numerical methods in a variety of practical matching problems. This paper studies a block-structured double-ended queue, whose block struct…
The intrinsic volumes induced by a stationary Poisson k-flat process inside a compact and convex sampling window are considered. Using techniques from stochastic analysis, more precisely calculus with multiple stochastic integrals and a Wiener-Ito chaos expansion of Poisson functionals, all moments and cumulants, exact and asymptotic, are calculated in terms of a family of integral-geometric functionals. Moreover, univariate central limit theorems as well as Berry-Esseen-type inequalities are shown and a multidimensional limit theorem is concluded.
Stationary stochastic processes with independent increments, of which the Poisson process is a prominent example, are widely used to describe real world events. With the basic assumption that a counting process is stationary and has independent increments, here I derive two integral equations to capture the time evolution of any such process. In order to solve these two integral equations explicitly, I need to introduce one more restriction condition. For sake of simplicity, I have imposed the Poisson condition on the two equations and successfully reproduced the renown Poisson results. The me…
We consider the Lévy model of the perpetual American call and put options with a negative discount rate under Poisson observations. Similar to the continuous observation case as in De Donno et al. [24], the stopping region that characterizes the optimal stopping time is either a half-line or an interval. The objective of this paper is to obtain explicit expressions of the stopping and continuation regions and the value function, focusing on spectrally positive and negative cases. To this end, we compute the identities related to the first Poisson arrival time to an interval via the scale funct…
In this paper, we study existence and uniqueness of strong as well as weak solutions for general time fractional Poisson equations. We show that there is an integral representation of the solutions of time fractional Poisson equations with zero initial values in terms of semigroup for the infinitesimal spatial generator ${\cal L}$ and the corresponding subordinator associated with the time fractional derivative. This integral representation has an integral kernel $q(t, x, y)$, which we call the fundamental solution for the time fractional Poisson equation, if the semigrou for ${\cal L}$ has an…
We prove necessary and sufficient conditions for the asymptotic normality of multiple integrals with respect to a Poisson measure on a general measure space, expressed both in terms of norms of contraction kernels and of variances of carré-du-champ operators. Our results substantially complete the fourth moment theorems recently obtained by Döbler and Peccati (2018) and Döbler, Vidotto and Zheng (2018). An important tool for achieving our goals is a novel product formula for multiple integrals under minimal conditions.
Expansion of Iterated Stochastic Integrals with Respect to Martingale Poisson Measures and with Respect to Martingales Based on Generalized Multiple Fourier Series
