Limit theorems for Markovian Hawkes processes with a large initial intensity

Hawkes process is a simple point process 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, social networks, criminology, seismology, and many other fields. In this paper, we study the linear Hawkes process with an exponential exciting function in the asymptotic regime where the initial intensity of the Hawkes process is large. We derive limit theorems under this asymptotic regime as well as the regime when both the initial intensity and the time are large.     

Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

A univariate Hawkes process is a simple point process that is self-exciting and has a 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 processes have 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 dependent increments. We use the resulting approximation to study an infinite-server queue with high-volume Hawkes traffic. We show that the queue length process can be approximated by a Gaussian process, for which we compute explicitly the covariance function and the steady-state distribution. We also extend our results to multivariate stationary Hawkes processes and establish limit theorems for infinite-server queues with multivariate Hawkes traffic.     

Hawkes process and Edgeworth expansion with application to maximum likelihood estimator

Statistical Inference for Stochastic Processes - We provide a rigorous mathematical foundation of the theory for the higher-order asymptotic behavior of the one-dimensional Hawkes process with an...     

Direct Likelihood Evaluation for the Renewal Hawkes Process

An interesting extension of the widely applied Hawkes self-exiting point process, the renewal Hawkes (RHawkes) process, was recently proposed by Wheatley, Filimonov, and Sornette, which has the potential to significantly widen the application domains of the self-exciting point processes. However, they claimed that computation of the likelihood of the RHawkes process requires exponential time and therefore is practically impossible. They proposed two expectation–maximization (EM) type algorithms to compute the maximum likelihood estimator (MLE) of the model parameters. Because of the fundamental role of likelihood in statistical inference, a practically feasible method for likelihood evaluation is highly desirable. In this article, we provide an algorithm that evaluates the likelihood of the RHawkes process in quadratic time, a drastic improvement from the exponential time claimed by Wheatley, Filimonov, and Sornette. We demonstrate the superior performance of the resulting MLEs of the model relative to the EM estimators through simulations. We also present a computationally efficient procedure to calculate the Rosenblatt residuals of the process for goodness-of-fit assessment, and a simple yet efficient procedure for future event prediction. The proposed methodologies were applied on real data from seismology and finance. An R package implementing the proposed methodologies is included in the supplementary materials.     

A risk model with paying dividends and random environment

We consider a discrete time risk model where dividends are paid to insureds and the claim size has a discrete phase-type distribution, but the claim sizes vary according to an underlying Markov process called an environment process. In addition, the probability of paying the next dividend is affected by the current state of the underlying Markov process. We provide explicit expressions for the ruin probability and the deficit distribution at ruin by extracting a QBD (quasi-birth-and-death) structure in the model and then analyzing the QBD process. Numerical examples are also given.     

Some asymptotic results for nonlinear Hawkes processes

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 on a large network.     

Mean-field limit of generalized Hawkes processes

We generalize multivariate Hawkes processes mainly by including a dependence with respect to the age of the process, i.e. the delay since the last point.Within this class, we investigate the limit behaviour, when $n$ goes to infinity, of a system of $n$ mean-field interacting age-dependent Hawkes processes. We prove that such a system can be approximated by independent and identically distributed age dependent point processes interacting with their own mean intensity. This result generalizes the study performed by Delattre et al. (2016).In continuity with Chevallier et al. (2015), the second goal of this paper is to give a proper link between these generalized Hawkes processes as microscopic models of individual neurons and the age-structured system of partial differential equations introduced by Pakdaman et al. (2010) as macroscopic model of neurons.     

稀疏过程下变破产下限多元相依风险模型的破产概率

研究了稀疏过程下多元相依风险模型在假定变破产下限的破产概率,其中索赔产生时依赖概率ρ的可能性同时产生一次续保,即续保过程是索赔的ρ-稀疏过程.运用鞅方法得到了当破产下限为某些特征函数时破产概率所满足的不等式或破产概率的具体表达式.     

Approximate Simulation of Hawkes Processes

Hawkes processes are important in point process theory and its applications, and simulation of such processes are often needed for various statistical purposes. This article concerns a simulation algorithm for unmarked and marked Hawkes processes, exploiting that the process can be constructed as a Poisson cluster process. The algorithm suffers from edge effects but is much faster than the perfect simulation algorithm introduced in our previous work Møller and Rasmussen (2004). We derive various useful measures for the error committed when using the algorithm, and we discuss various empirical results for the algorithm compared with perfect simulations. Extensions of the algorithm and the results to more general types of marked point processes are also discussed.     

Bayesian Inference for Hawkes Processes

The Hawkes process is a practically and theoretically important class of point processes, but parameter-estimation for such a process can pose various problems. In this paper we explore and compare two approaches to Bayesian inference. The first approach is based on the so-called conditional intensity function, while the second approach is based on an underlying clustering and branching structure in the Hawkes process. For practical use, MCMC (Markov chain Monte Carlo) methods are employed. The two approaches are compared numerically using three examples of the Hawkes process.     

Moderate deviations for marked Hawkes processes

We consider a linear Hawkes process with random marks. Some limit theorems have been studied by Karabash and Zhu [Stoch. Models, 31, 433–451 (2015)]. In this paper, we obtain a moderate deviation principle for marked Hawkes processes.     

A class of two-sex branching processes with reproduction phase in a random environment

We model the demographic dynamics of populations with sexual reproduction where the reproduction phase occurs in a non-predictable environment and we assume the immigration/out-migration of mating units in the population. We introduce a general class of two-sex branching processes where, in each generation, the number of mating units which take part in the reproduction phase is randomly determined and the offspring probability distribution changes over time in a random environment. We provide several probabilistic results about the limit behaviour of populations whose dynamics is modelled by such a class of stochastic processes. In particular, we provide sufficient conditions for the almost sure extinction of the population or for its survival with a positive probability. As illustration, we include some simulated examples.     

Joint survival probability via truncated invariant copula

Given an intensity-based credit risk model, this paper studies dependence structure between default intensities. To model this structure, we use a multivariate shot noise intensity process, where jumps occur simultaneously and their sizes are correlated. Through very lengthy algebra, we obtain explicitly the joint survival probability of the integrated intensities by using the truncated invariant Farlie–Gumbel–Morgenstern copula with exponential marginal distributions. We also apply our theoretical result to pricing basket default swap spreads. This result can provide a useful guide for credit risk management.     

Delay analysis of a two-class batch-service queue with class-dependent variable server capacity

In this paper, we analyse the delay of a random customer in a two-class batch-service queueing model with variable server capacity, where all customers are accommodated in a common single-server first-come-first-served queue. The server can only process customers that belong to the same class, so that the size of a batch is determined by the length of a sequence of same-class customers. This type of batch server can be found in telecommunications systems and production environments. We first determine the steady state partial probability generating function of the queue occupancy at customer arrival epochs. Using a spectral decomposition technique, we obtain the steady state probability generating function of the delay of a random customer. We also show that the distribution of the delay of a random customer corresponds to a phase-type distribution. Finally, some numerical examples are given that provide further insight in the impact of asymmetry and variance in the arrival process on the number of customers in the system and the delay of a random customer.     

Asymptotic probability of majority inversion under a general binomial voting model

We obtain limits on the probability of majority inversion when the number of voters tends to infinity, for a binomial voting model specific to each state with different population sizes, and for different voting quotas in the two stages of the voting procedure. The case of weighted votes at the second stage is also discussed. For an important special case where the limit cannot be determined, we provide an exact expression for the inversion probability, but only for unweighted votes.     

广义负相依一般风险模型中有限时破产概率的估计及数值模拟

本文研究了一类带利率的重尾相依风险模型, 其中索赔额是一列上广义负相依随机变量, 索赔到达过程是一般的非负整值过程, 并且独立于索赔额序列, 保费收入过程是一个一般的非负非降随机过程. 我们考虑了两种情况, 其一是索赔额、索赔到达过程及保费收入过程相互独立, 其二是累积折现保费收入总量的尾概率可以被索赔额的尾概率高阶控制, 得到了保险公司有限时破产概率的渐近估计,并且给出了相应的数值模拟, 验证了理论结果的合理性.     

Markov modulated Bernoulli process

We consider a Bernoulli process where the success probability changes with respect to a Markov chain. Such a model represents an interesting application of stochastic processes where the parameters are not constants; rather, they are stochastic processes themselves due to their dependence on a randomly changing environment. The model operates in a random environment depicted by a Markov chain so that the probability of success at each trial depends on the state of the environment. We will concentrate, in particular, on applications in reliability theory to motivate our model. The analysis will focus on transient as well as long-term behaviour of various processes involved.     

Risk Processes with Non-stationary Hawkes Claims Arrivals

We consider risk processes with non-stationary Hawkes claims arrivals, and we study the asymptotic behavior of infinite and finite horizon ruin probabilities under light-tailed conditions on the claims. Moreover, we provide asymptotically efficient simulation laws for ruin probabilities and we give numerical illustrations of the theoretical results.     

On the asymptotics of fault probability in least‐recently‐used caching with Zipf‐type request distribution

We consider the least‐recently‐used cache replacement rule with a Zipf‐type page request distribution and investigate an asymptotic property of the fault probability with respect to an increase of cache size. We first derive the asymptotics of the fault probability for the independent‐request model and then extend this derivation to a general dependent‐request model, where our result shows that under some weak assumptions the fault probability is asymptotically invariant with regard to dependence in the page request process. In a previous study, a similar result was derived by applying a Poisson embedding technique, where a continuous‐time proof was given through some assumptions based on a continuous‐time modeling. The Poisson embedding, however, is just a technique used for the proof and the problem is essentially on a discrete‐time basis; thus, it is preferable to make assumptions, if any, directly in the discrete‐time setting. We consider a general dependent‐request model and give a direct discrete‐time proof under different assumptions. A key to the proof is that the numbers of requests for respective pages represent conditionally negatively associated random variables. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Concentration inequalities for stochastic differential equations of pure non-Poissonian jumps

We provide concentration inequalities for solutions to stochastic differential equations of pure not-necessarily Poissonian jumps. Our proofs are based on transportation cost inequalities for square integrable functionals of point processes with stochastic intensity and elements of stochastic calculus with respect to semi-martingales. We apply the general results to solutions of stochastic differential equations driven by renewal and non-linear Hawkes point processes.