In this paper, we consider an extension of the Poisson random measure for the formulation of continuous-time reinforcement learning, such that both the frequency and the width of the jumps depend on the path. Starting from a general point process, we define a new Poisson random measure as limit of the linear sum of these counting processes, and name it the Mesgaki random measure. We also construct its Stochastic integral and Itô's formula.
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.
We construct a family of processes, from a single Poisson process, that converges in law to a complex Brownian motion. Moreover, we find realizations of these processes that converge almost surely to the complex Brownian motion, uniformly on the unit time interval. Finally the rate of convergence is derived.
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…
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 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.
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 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
This article employs the relation between probabilities of two consecutive values of a Poisson random variable to derive conditions for the weak convergence of point processes to a Poisson process. As applications, we consider the starting points of k-runs in a sequence of Bernoulli random variables and point processes constructed using inradii and circumscribed radii of inhomogeneous Poisson-Voronoi tessellations.
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.
Dynamical scaling is an asymptotic property typical for the dynamics of first-order phase transitions in physical systems and related to self-similarity. Based on the integral-representation for the marginal probabilities of a fractional non-homogeneous Poisson process introduced by Leonenko et al. (2017) and generalising the standard fractional Poisson process, we prove the dynamical scaling under fairly mild conditions. Our result also includes the special case of the standard fractional Poisson process.
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…
The simple Lévy Poisson process and scaled forms are explicitly constructed from partial sums of independent and identically distributed random variables and from sums of non-stationary independent random variables. For the latter, the weak limits are scaled Poisson processes. The method proposed here prepares generalizations to dependent data, to associated data in the first place.
We give Hardy-Stein and Douglas identities for nonlinear nonlocal Sobolev-Bregman integral forms with unimodal Lévy measures. We prove that the corresponding Poisson integral defines an extension operator for the Sobolev-Bregman spaces. We also show that the Poisson integrals are quasiminimizers of the Sobolev-Bregman forms.
Moore and Yang defined an integral notion, based on an extension of Riemann sums, for inclusion monotonic continuous interval functions, where the integrations limits are real numbers. This integral notion extend the usual integration of real functions based on Riemann sums. In this paper, we extend this approach by considering intervals as integration limits instead of real numbers and we abolish the inclusion monotonicity restriction of the interval functions and this notion is used to determine interval probability density functions.
