Giovanni Luca Torrisi
We give a general Gaussian bound for the first chaos (or innovation) of point processes with stochastic intensity constructed by embedding in a bivariate Poisson process. We apply the general result to nonlinear Hawkes processes, providing quantitative central limit theorems.
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Hawkes process is a class of simple point processes that is self-exciting and has clustering effect. The intensity of this point process depends on its entire past history. It has wide applications in finance, neuroscience and many other fields. In this paper, we study the large deviations for nonlinear Hawkes processes. The large deviations for linear Hawkes processes has been studied by Bordenave and Torrisi. In this paper, we prove first a large deviation principle for a special class of nonlinear Hawkes processes, that is, a Markovian Hawkes process with nonlinear rate and exponential exci…
Hawkes process is a class of simple point processes with self-exciting and clustering properties. Hawkes process has been widely applied in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we study fluctuations, large deviations and moderate deviations nonlinear Hawkes processes in a new asymptotic regime, the large intensity function and the small exciting function regime. It corresponds to the large baseline intensity asymptotics for the linear case, and can also be interpreted as the asymptotics for the mean process of Hawkes processes o…
Hawkes process is a self-exciting point process with clustering effect whose intensity depends on its entire past history. It has wide applications in neuroscience, finance and many other fields. In this paper, we obtain a functional central limit theorem for nonlinear Hawkes process. Under the same assumptions, we also obtain a Strassen's invariance principle, i.e. a functional law of the iterated logarithm.
In this paper, we prove a process-level, also known as level-3 large deviation principle for a very general class of simple point processes, i.e. nonlinear Hawkes process, with a rate function given by the process-level entropy, which has an explicit formula.
The Hawkes process is a simple point process that has long memory, clustering effect, self-exciting property and is in general non-Markovian. The future evolution of a self-exciting point process is influenced by the timing of the past events. There are applications in finance, neuroscience, genome analysis, seismology, sociology, criminology and many other fields. We first survey the known results about the theory and applications of both linear and nonlinear Hawkes processes. Then, we obtain the central limit theorem and process-level, i.e. level-3 large deviations for nonlinear Hawkes proce…
We consider spatially extended systems of interacting nonlinear Hawkes processes modeling large systems of neurons placed in Rd and study the associated mean field limits. As the total number of neurons tends to infinity, we prove that the evolution of a typical neuron, attached to a given spatial position, can be described by a nonlinear limit differential equation driven by a Poisson random measure. The limit process is described by a neural field equation. As a consequence, we provide a rigorous derivation of the neural field equation based on a thorough mean field analysis.
Gaussian approximation for the integrated intensity of counting processes: application to Hawkes processes
本文针对计数过程强度的积分波动建立定量中心极限定理，推导高斯随机变量与中心化强度归一化积分的Wasserstein距离上界，并将结果应用于次临界线性Hawkes过程，证明其积分波动依分布收敛于中心化高斯随机变量。
Moderate Deviations for Nonlinear Hawkes Processes
本文为非线性霍克斯过程建立完全适度区间内的样本路径适度偏差原理，通过构造泊松校正子并利用鞅方法，得到介于中心极限与大偏差尺度间的结果，并给出渐近方差响应公式。
Gaussian Approximation and Moderate Deviations of Poisson Shot Noises with Application to Compound Generalized Hawkes Processes
本文研究了泊松脉冲噪声的高斯近似与中等偏差，推导了相关不等式，并应用于广义复合Hawkes过程。
We investigate the asymptotic behaviour of networks of interacting non-linear Hawkes processes modeling a homogeneous population of neurons in the large population limit. In particular, we prove a functional central limit theorem for the mean spike-activity thereby characterizing the asymptotic fluctuations in terms of a stochastic Volterra integral equation. Our approach differs from previous approaches in making use of the associated resolvent in order to represent the fluctutations as Skorokhod continuous mappings of weakly converging martingales. Since the Lipschitz properties of the resol…
A univariate Hawkes process is a simple point process that is self-exciting and has clustering effect. The intensity of this point process is given by the sum of a baseline intensity and another term that depends on the entire past history of the point process. Hawkes process has wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we prove a functional central limit theorem for stationary Hawkes processes in the asymptotic regime where the baseline intensity is large. The limit is a non-Markovian Gaussian process with depe…
In this paper, following Nourdin-Peccati's methodology, we combine the Malliavin calculus and Stein's method to provide general bounds on the Wasserstein distance between functionals of a compound Hawkes process and a given Gaussian density. To achieve this, we rely on the Poisson embedding representation of an Hawkes process to provide a Malliavin calculus for the Hawkes processes, and more generally for compound Hawkes processes. As an application, we close a gap in the literature by providing the first Berry-Esséen bounds associated to Central Limit Theorems for the compound Hawkes process.
In this paper, we provide upper bounds on the d2 distance between a large class of functionals of a multivariate compound Hawkes process and a given Gaussian vector. This is proven using Malliavin's calculus defined on an underlying Poisson embedding. The upper bound is then used to infer the speed of convergence of Central Limit Theorems for the multivariate compound Hawkes process with exponential kernels as the observation time T goes to infinity.
Normal approximation of Functionals of Point Processes: Application to Hawkes Processes
本文利用Stein方法、Malliavin演算和Poisson嵌入表示，给出了点过程泛函与高斯分布之间Wasserstein距离的显式上界，并应用于离散、连续及近不稳定Hawkes过程，建立了显式收敛速率。
This paper establishes a functional law of large numbers and a functional central limit theorem for marked Hawkes point measures and their corresponding shot noise processes. We prove that the normalized random measure can be approximated in distribution by the sum of a Gaussian white noise process plus an appropriate lifting map of a correlated one-dimensional Brownian motion. The Brownian results from the self-exiting arrivals of events. We apply our limit theorems for Hawkes point measures to analyze the population dynamics of budding microbes in a host.
In the last decade Hawkes processes have received much attention as models for functional connectivity in neural spiking networks and other dynamical systems with a cascade behavior. In this paper we establish a renewal approach for analyzing this process. We consider the ordinary nonlinear Hawkes process as well as the more recently described age dependent Hawkes process. We construct renewal-times and establish moment results for these. This gives rise to study the Hawkes process as a Markov chain. As an application, we prove asymptotic results such as a functional CLT and a time-average CLT…
We consider a classical risk process with arrival of claims following a non-stationary Hawkes process. We study the asymptotic regime when the premium rate and the baseline intensity of the claims arrival process are large, and claim size is small. The main goal of the article is to establish a diffusion approximation by verifying a functional central limit theorem and to compute the ruin probability in finite-time horizon. Numerical results will also be given.
Large deviations for the mean-field limit of Hawkes processes
研究多维非线性Hawkes过程在维数趋于无穷时的平均场极限，证明其满足大偏差原理。
Malliavin calculcus for a Hawkes process
本文为非线性Hawkes过程建立Malliavin微积分，通过扰动跳跃时间构造局部Dirichlet型，并据此给出相关SDE解绝对连续性的判据及金融衍生品敏感性公式。
Local stability and rates of convergence to equilibrium for the Nonlinear Renewal Equation; applications to Hawkes processes
本文研究非线性再生方程的局部稳定性及收敛速率，应用于Hawkes过程的中心极限定理推导。
Fluctuations and moderate deviations for the mean fields of Hawkes processes
本文研究了多元非线性 Hawkes 过程均场的涨落及其相关的大偏差（即中偏差），补充了均场极限为时间非齐次 Poisson 过程后的精细化极限理论。
We propose a mean-field model of interacting point processes where each process has a memory of the time elapsed since its last event (age) and its recent past (leaky memory), generalizing Age-dependent Hawkes processes. The model is motivated by interacting nonlinear Hawkes processes with Markovian self-interaction and networks of spiking neurons with adaptation and short-term synaptic plasticity. By proving propagation of chaos and using a path integral representation for the law of the limit process, we show that, in the mean-field limit, the empirical measure of the system follows a multid…
The propagation of chaos and associated law of large numbers for mean-field interacting age-dependent Hawkes processes (when the number of processes n goes to +$\infty$) being granted by the study performed in (Chevallier, 2015), the aim of the present paper is to prove the resulting functional central limit theorem. It involves the study of a measure-valued process describing the fluctuations (at scale n --1/2) of the empirical measure of the ages around its limit value. This fluctuation process is proved to converge towards a limit process characterized by a limit system of stochastic differ…
We consider a nonlinear multivariate Hawkes process having a variable length memory which allows to describe the activity of a neuronal network by its membrane potential. We propose a graphical construction of the process and we construct, by means of a perfect simulation algorithm, a stationary version of the process. By making the hypothesis that the spiking rate $β_i$ of the neuron $i \in I $ is bounded, we construct an algorithm based on a priori realizations of the Poisson process $(M^i, i \in I)$. We show that there exists a critical value $δ_c$ such that if $\underlineδ > δ_c$ (where $\…
Functional limiting behaviour of non-stationary marked Hawkes processes under multi-scaling high-intensity regime
本文研究非平稳标记霍克斯过程在高强度机制下的函数极限行为，建立了函数大数定律和中心极限定理，证明重标度过程收敛于高斯过程，并研究了散粒噪声过程。
Loglinear Hawkes processes: existence, nonexplosion, stability and moment bounds
本文研究速率函数为指数函数的对数线性 Hawkes 过程，给出了爆炸与非爆炸的充分条件，并证明非正记忆函数下过程具有稳定性，为后续研究提供理论基础。
Convergence of nonhomogeneous Hawkes processes and Feller random measures
研究非齐次Hawkes过程在近不稳定极限区域的缩放极限，其激发测度可能依赖于世代，通过非线性卷积方程刻画极限随机测度，扩展了Feller扩散和分数Feller（CIR）过程，并允许一般Levy型算子的驱动噪声。
A Hawkes Microfoundation for Multitype Inverse Gaussian Subordinators
本文为多类型逆高斯子序提供事件级Hawkes微观基础，证明近临界多变量Hawkes过程收敛到该子序，并揭示其机制与标度理论关系。
In a previous paper, it has been shown that the mean-field limit of spatially extended Hawkes processes is characterized as the unique solution $u(t,x)$ of a neural field equation (NFE). The value $u(t,x)$ represents the membrane potential at time $t$ of a typical neuron located in position $x$, embedded in an infinite network of neurons. In the present paper, we complement this result by studying the fluctuations of such a stochastic system around its mean-field limit $u(t,x)$. Our first main result is a central limit theorem stating that the spatial distribution associated with these fluctua…
Efficient Simulation of Hawkes Processes using their Affine Volterra Structure
针对Hawkes过程，提出基于仿射Volterra结构的固定时间网格高效模拟方案，先模拟积分强度与计数过程再恢复跳跃时间，具确定性复杂度，适用于各类核函数，弱收敛性获证且数值实验显示精度高、效率优。
A generalized central limit theorem for critical marked Hawkes processes
本文证明了临界标记霍克斯过程的中心极限定理，针对重尾或有限方差标记，得到稳定过程或高斯过程极限，并提出了适用于更广泛自激系统的新方法。
In the present work, we introduce a general class of mean-field interacting nonlinear Hawkes processes modelling the reciprocal interactions between two neuronal populations, one excitatory and one inhibitory. The model incorporates two features: inhibition, which acts as a multiplicative factor onto the intensity of the excitatory population and additive retroaction from the excitatory neurons onto the inhibitory ones. We give first a detailed analysis of the well-posedness of this interacting system as well as its dynamics in large population. The second aim of the paper is to give a rigorou…
Perfect simulation for interacting Hawkes processes with reset-induced variable length memory
本文研究了具有重置机制的相互作用非线性Hawkes过程，通过图构造和祖先族方法，证明了子临界条件下的有限祖先族，从而构建了平稳解并提出精确的完美模拟算法。
The problem of finding a necessary and sufficient condition for the continuity of the local times for a general Markov process is still open. Barlow and Hawkes have completely treated the case of the Lévy processes, and Marcus and Rosen have solved the case of the strongly symmetric Markov processes. We treat here the continuity of the local times of Borel right processes. Our approach unifies that of Barlow and Hawkes and of Marcus and Rosen, by using an associated Gaussian process, that appears as a limit in a CLT involving the local time process.
We consider multi-class systems of interacting nonlinear Hawkes processes modeling several large families of neurons and study their mean field limits. As the total number of neurons goes to infinity we prove that the evolution within each class can be described by a nonlinear limit differential equation driven by a Poisson random measure, and state associated central limit theorems. We study situations in which the limit system exhibits oscillatory behavior, and relate the results to certain piecewise deterministic Markov processes and their diffusion approximations.
Population equations for infinitely large networks of spiking neurons have a long tradition in theoretical neuroscience. In this work, we analyze a recent generalization of these equations to populations of finite size, which takes the form of a nonlinear stochastic integral equation. We prove that, in the case of leaky integrate-and-fire (LIF) neurons with escape noise and for a slightly simplified version of the model, the equation is well-posed and stable in the sense of Brémaud-Massoulié. The proof combines methods from Markov processes taking values in the space of positive measures and n…
The limit Gaussian distribution of multivariate weighted functionals of nonlinear transformations of Gaussian stationary processes, having multiple singular spectra, is derived, under very general conditions on the weight function. This paper is motivated by its potential applications in nonlinear regression, and asymptotic inference on nonlinear functionals of Gaussian stationary processes with singular spectra.
We prove a limit theorem on the convergence of the distributions of the scaled last exit time over a slowly moving nonlinear boundary for a class of Gaussian stationary processes. The limit is a double exponential (Gumbel) distribution.
We present a self-contained proof of a uniform bound on multi-point correlations of trigonometric functions of a class of Gaussian random fields. It corresponds to a special case of the general situation considered in [Hairer-Xu], but with improved estimates. As a consequence, we establish convergence of a class of Gaussian fields composite with more general functions. These bounds and convergences are useful ingredients to establish weak universalities of several singular stochastic PDEs.
Gaussian fluctuations of a nonlinear stochastic heat equation in dimension two
研究二维非线性随机热方程在乘性高斯噪声下的高斯涨落，证明经中心化和重标度后解随机场收敛至Edwards-Wilkinson极限，并给出函数版本结果。
Approximating the solution of the nonlinear filtering problem with Gaussian mixtures has been a very popular method since the 1970s. However, the vast majority of such approximations are introduced in an ad-hoc manner without theoretical grounding. This work is a continuation of [4, 5], where we described a rigorous Gaussian mixture approximation to the solution of the filtering problem. We deduce here a refined estimate of the rate of convergence of the approximation. We do this by proving a central limit type theorem for the error process. We also find the optimal variances of the Gaussian m…
Uniform-in-time Gaussian fluctuations for multiscale nonlinear stochastic systems via Malliavin Calculus
利用Malliavin微分的二阶Poincaré不等式，在较弱条件下建立了非线性慢快随机系统波动过程与高斯随机变量之间Wasserstein距离的一致时间界。
We show how it is possible to assess the rate of convergence in the Gaussian approximation of triangular arrays of $U$-statistics, built from wavelets coefficients evaluated on a homogeneous spherical Poisson field of arbitrary dimension. For this purpose, we exploit the Stein-Malliavin approach introduced in the seminal paper by Peccati, Solé, Taqqu and Utzet (2011); we focus in particular on statistical applications covering evaluation of variance in non-parametric density estimation and Sobolev tests for uniformity.
We consider the cubic fourth order nonlinear Schrödinger equation on the circle. In particular, we prove that the mean-zero Gaussian measures on Sobolev spaces $H^s(\mathbb{T})$, $s > \frac34$, are quasi-invariant under the flow.
Under certain regularity conditions, we establish quasi-invariance of Gaussian measures on periodic functions under the flow of cubic fractional nonlinear Schrödinger equations on the one-dimensional torus.
We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the two-dimensional defocusing cubic nonlinear wave equation (NLW). Under some regularity condition, we prove quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces for the NLW dynamics. We achieve this goal by introducing a simultaneous renormalization on the energy functional and its time derivative and establishing a renormalized energy estimate in the probabilistic setting.
We prove quasi-invariance of Gaussian measures $μ_s$ with Cameron-Martin space $H^s$ under the flow of the defocusing nonlinear wave equation with polynomial nonlinearities of any order for all $s>5/2$, including fractional $s$. This extends work of Oh-Tzvetkov [15] and Gunaratnam-Oh-Tzvetkov-Weber [10] who proved this result this result for a cubic nonlinearity and $s$ an even integer. The main contributions are a modified construction of a weighted adapted to higher order nonlinearity, and an energy estimate for the derivative of the energy replacing the integration by parts argument introdu…
We continue the study on the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation. By considering the renormalized equation, we extend the quasi-invariance results in [30, 27] to Sobolev spaces of negative regularity. Our proof combines the approach introduced by Planchon, Tzvetkov, and Visciglia [35] with the normal form approach in [30, 27].
We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation on the circle. In particular, we establish an optimal regularity result for quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces. The main new ingredient is an improved energy estimate established by performing an infinite iteration of normal form reductions on the energy functional. Furthermore, we show that the dispersion is essential for such a quasi-invariance result by proving non quasi-invariance of the Gaussian measures…
We prove quasi-invariance of Gaussian measures supported on Sobolev spaces under the dynamics of the three-dimensional defocusing cubic nonlinear wave equation. As in the previous work on the two-dimensional case, we employ a simultaneous renormalization on the energy functional and its time derivative. Two new ingredients in the three-dimensional case are (i) the construction of the weighted Gaussian measures, based on a variational formula for the partition function inspired by Barashkov and Gubinelli (2018), and (ii) an improved argument in controlling the growth of the truncated weighted G…
We introduce a nonlinear modification of the classical Hawkes process, which allows inhibitory couplings between units without restrictions. The resulting system of interacting point processes provides a useful mathematical model for recurrent networks of spiking neurons with exponential transfer functions. The expected rates of all neurons in the network are approximated by a first-order differential system. We study the stability of the solutions of this equation, and use the new formalism to implement a winner-takes-all network that operates robustly for a wide range of parameters. Finally,…
Quasi-invariance of Gaussian measures for the $3d$ energy critical nonlinear Schr\" odinger equation
本文研究三维能量临界非线性薛定谔方程在高斯测度下的演化，证明当协方差算子$(1-\Delta)^{-s}$中$s$足够大时，方程流保持满测度集的拟不变性，从而将已有的一维结果推广到高维情形。
Performance guarantees for compression in nonlinear models under non-Gaussian observations can be achieved through the use of distributional characteristics that are sensitive to the distance to normality, and which in particular return the value of zero under Gaussian or linear sensing. The use of these characteristics, or discrepancies, improves some previous results in this area by relaxing conditions and tightening performance bounds. In addition, these characteristics are tractable to compute when Gaussian sensing is corrupted by either additive errors or mixing.
We study the transport property of Gaussian measures on Sobolev spaces of periodic functions under the dynamics of the one-dimensional cubic fractional nonlinear Schrödinger equation. For the case of second-order dispersion or greater, we establish an optimal regularity result for the quasi-invariance of these Gaussian measures, following the approach by Debussche and Tsutsumi [15]. Moreover, we obtain an explicit formula for the Radon-Nikodym derivative and, as a corollary, a formula for the two-point function arising in wave turbulence theory. We also obtain improved regularity results in th…
Quasi-invariance of the Gaussian measure for the two-dimensional stochastic cubic nonlinear wave equation
本文研究二维环面上带时空白噪声的随机阻尼三次非线性波动方程，证明了对应线性方程唯一不变测度的高斯测度在非线性随机流下具有拟不变性。
We study the ensemble Kalman filter (EnKF) algorithm for sequential data assimilation in a general situation, that is, for nonlinear forecast and measurement models with non-additive and non-Gaussian noises. Such applications traditionally force us to choose between inaccurate Gaussian assumptions that permit efficient algorithms (e.g., EnKF), or more accurate direct sampling methods which scale poorly with dimension (e.g., particle filters, or PF). We introduce a trimmed ensemble Kalman filter (TEnKF) which can interpolate between the limiting distributions of the EnKF and PF to facilitate ad…
Rethinking Nonlinearity: Trainable Gaussian Mixture Modules for Modern Neural Architectures
本文提出高斯混合启发式非线性模块（GMNM），利用GMM密度逼近与高斯核距离属性，可灵活集成于MLP、CNN、注意力及LSTM等架构，端到端训练并持续提升性能。
We consider the Cauchy problem for the fractional nonlinear Schrödinger equation (FNLS) on the one-dimensional torus with cubic nonlinearity and high dispersion parameter $α> 1$, subject to a Gaussian random initial data of negative Sobolev regularity $σ<s-\frac{1}{2}$, for $s \le \frac 12$. We show that for all $s_{*}(α) <s\leq \frac{1}{2}$, the equation is almost surely globally well-posed. Moreover, the associated Gaussian measure supported on $H^{s}(\mathbb T)$ is quasi-invariant under the flow of the equation. For $α< \frac{1}{20}(17 + 3\sqrt{21}) \approx 1.537$, the regularity of the ini…
The main purpose of this paper is to study the problem of determining initial condition of nonlinear parabolic equation from noisy observations of the final condition. We introduce a regularized method to establish an approximate solution. We prove an upper bound on the rate of convergence of the mean integrated squared error.
The Fokker-Planck equations describe time evolution of probability densities of stochastic dynamical systems and are thus widely used to quantify random phenomena such as uncertainty propagation. For dynamical systems driven by non-Gaussian Lévy processes, however, it is difficult to obtain explicit forms of Fokker-Planck equations because the adjoint operators of the associated infinitesimal generators usually do not have exact formulation. In the present paper, Fokker- Planck equations are derived in terms of infinite series for nonlinear stochastic differential equations with non-Gaussian L…
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