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We introduce a new point process, the dynamic contagion process, by generalising the Hawkes process and the Cox process with shot noise intensity. Our process includes both self-excited and externally excited jumps, which could be used to model the dynamic contagion impact from endogenous and exogenous factors of the underlying system. We have systematically analysed the theoretical distributional properties of this new process, based on the piecewise-deterministic Markov process theory developed in Davis (1984), and the extension of the martingale methodology used in Dassios and Jang (2003). The analytic expressions of the Laplace transform of the intensity process and the probability generating function of the point process have been derived. An explicit example of specified jumps with exponential distributions is also given. The object of this study is to produce a general mathematical framework for modelling the dependence structure of arriving events with dynamic contagion, which has the potential to be applicable to a variety of problems in economics, finance, and insurance. We provide an application of this process to credit risk, and a simulation algorithm for further industrial implementation and statistical analysis.

This tutorial provides an introduction to Palm distributions for spatial point processes. Initially, in the context of finite point processes, we give an explicit definition of Palm distributions in terms of their density functions. Then we review Palm distributions in the general case. Finally, we discuss some examples of Palm distributions for specific models and some applications.

We consider spatial point processes with a pair correlation function, which depends only on the lag vector between a pair of points. Our interest is in statistical models with a special kind of 'structured' anisotropy : the pair correlation function is geometric anisotropic if it is elliptical but not spherical. In particular, we study Cox process models with an elliptical pair correlation function, including shot noise Cox processes and log Gaussian Cox processes, and we develop estimation procedures using summary statistics and Bayesian methods. Our methodology is illustrated on real and synthetic datasets of spatial point patterns.

Consider a multidimensional risk model, in which an insurer simultaneously confronts m (m ≥ 2) types of claims sharing a common non-stationary and non-renewal arrival process. Assuming that the claims arrival process satisfies a large deviation principle and the claim-size distributions are heavy-tailed, asymptotic estimates for two common types of ruin probabilities for this multidimensional risk model are obtained. As applications, we give two examples of the non-stationary point process: a Hawkes process and a Cox process with shot noise intensity, and asymptotic ruin probabilities are obtained for these two examples.

We summarize and discuss the current state of spatial point process theory and directions for future research, making an analogy with generalized linear models and random effect models, and illustrating the theory with various examples of applications. In particular, we consider Poisson, Gibbs and Cox process models, diagnostic tools and model checking, Markov chain Monte Carlo algorithms, computational methods for likelihood-based inference, and quick non-likelihood approaches to inference.

Spatio-temporal Cox point process models with a multiplicative structure for the driving random intensity, incorporating covariate information into temporal and spatial components, and with a residual term modelled by a shot-noise process, are considered. Such models are flexible and tractable for statistical analysis, using spatio-temporal versions of intensity and inhomogeneous K-functions, quick estimation procedures based on composite likelihoods and minimum contrast estimation, and easy simulation techniques. These advantages are demonstrated in connection with the analysis of a relatively large data set consisting of 2796 days and 5834 spatial locations of fires. The model is compared with a spatio-temporal log-Gaussian Cox point process model, and likelihood-based methods are discussed to some extent.

Marked doubly stochastic Poisson processes are a particular type of marked point processes that are characterized by the number of events in any time interval as being conditionally Poisson distributed, given another positive stochastic process called intensity. Here we consider a subclass of these processes in which the intensity is assumed to be a deterministic function of another nonexplosive marked point process. In particular, we will investigate an intensity jump process with an exponential decay having an analytic form for the distribution of the times and sizes of the jumps, which can be seen as a generalization of the classical shot noise process. Assuming that the intensity is unobservable, interest here is in its filtering, that is, in the computation of its conditional distribution, over a whole time interval, given an observed trajectory of realized events. Because, in general, this computation cannot be performed analytically, we propose a simulation method that provides an approximate solution, which relies on the reversible-jump Markov chain Monte Carlo algorithm. Interestingly, the proposed filtering algorithm also allows the setup of a likelihood-based procedure for the estimation of the parameters of the model based on stochastic versions of the expectation-maximization (EM) algorithm. The potential of the filtering and estimation methods proposed are illustrated through some simulation experiments as well as on a financial ultra-high-frequency dataset of intraday S&P500 futures prices.