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Crime has both varying patterns in space, related to features of the environment, economy and policing, and patterns in time arising from criminal behaviour, such as retaliation. Serious crimes may also be presaged by minor crimes of disorder. We demonstrate that these spatial and temporal patterns are generally confounded, requiring analyses to take both into account, and propose a spatiotemporal self-exciting point process model that incorporates spatial features, near repeat and retaliation effects, and triggering. We develop inference methods and diagnostic tools, such as residual maps, for this model, and through extensive simulation and crime data obtained from Pittsburgh, Pennsylvania, demonstrate its properties and usefulness.

We introduce a broad family of generalised self-exciting point processes with CIR-type intensities, and we develop associated algorithms for their exact simulation. The underlying models are extensions of the classical Hawkes process, which already has numerous applications in modelling the arrival of events with clustering or contagion effect in finance, economics, and many other fields. Interestingly, we find that the CIR-type intensity, together with its point process, can be sequentially decomposed into simple random variables, which immediately leads to a very efficient simulation scheme. Our algorithms are also pretty accurate and flexible. They can be easily extended to further incorporate externally excited jumps, or, to a multidimensional framework. Some typical numerical examples and comparisons with other well-known schemes are reported in detail. In addition, a simple application for modelling a portfolio loss process is presented. The online appendix is available at https://doi.org/10.1287/opre.2017.1640

Modelling spatial and temporal patterns in ecology is imperative to understand the complex processes inherent in ecological phenomena. Log‐Gaussian Cox processes are a popular choice among ecologists to describe the spatiotemporal distribution of point‐referenced data. In addition, point pattern models where events instigate others nearby (i.e., self‐exciting behaviour) are becoming increasingly popular to infer the contagious nature of events (e.g., animal sightings). While there are existing R packages that facilitate fitting spatiotemporal point processes and, separately, self‐exciting models, none incorporate both. We present an R package, stelfi, that fits spatiotemporal self‐exciting and log‐Gaussian Cox process models using Template Model Builder through a range of custom‐written C++ templates. We illustrate the use of stelfi's functions fitting models to Sasquatch (bigfoot) sightings data within the USA. The structure of these data is typical of many seen in ecology studies. We show, from a temporal Hawkes process to a spatiotemporal self‐exciting model, how the models offered by the package enable additional insights into the temporal and spatial progression of point pattern data. We present extensions to these well‐known models that include spatiotemporal self‐excitation and joint likelihood models, which are better suited to capture the complex mechanisms inherent in many ecological data. The package stelfi offers user‐friendly functionality, is open source, and is available from CRAN. It offers the implementation of complex spatiotemporal point process models in R for applications even beyond the field of ecology. We introduce the R package stelfi, available from the Comprehensive R Archive Network. This package allows users to fit temporal self‐exciting Hawkes models, spatial and spatiotemporal log‐Gaussian Cox process models and self‐exciting spatiotemporal models. The functionality of stelfi is illustrated using Sasquatch (bigfoot) sightings data shipped with the package.

A Hawkes process is also known under the name of a self-exciting point process and has numerous applications throughout science and engineering. We derive the statistical estimation (maximum likelihood estimation) and goodness-of-fit (mainly graphical) for multivariate Hawkes processes with possibly dependent marks. As an application, we analyze two data sets from finance.

The threshold Ornstein-Uhlenbeck process is a continuous-time threshold autoregressive process. It follows the Ornstein-Uhlenbeck dynamics when above or below a fixed threshold, but its coefficients can be discontinuous at the threshold. We discuss (quasi)-maximum likelihood estimation of the drift parameters, assuming continuous and discrete time observations. In the ergodic case, we derive the consistency and the speed of convergence of these estimators in long time and high frequency. Based on these results, we develop a test for the presence of a threshold in the dynamics. Finally, we apply these statistical tools to short-term US interest rates modeling.

Mean reversion, stochastic volatility, convenience yield and presence of jump clustering are well documented salient features of commodity markets, where Asian options are very popular. We propose a model which takes into account all these stylized features. We first state our model under the historical measure, then, after introducing a structure preserving change of measure, we provide a risk-neutral version of the same model and we show how to price geometric and arithmetic Asian options. To this end, we derive semi-closed formulas for the geometric Asian options price and develop a computationally efficient simulation scheme for the price process, allowing to price the arithmetic counterparts using control variate technique. Finally, we propose a simple econometric experiment to document presence of jump clusters in commodity prices and evaluate the performances of the proposed simulation scheme on some parameter sets calibrated on real data.

High-dimensional self-exciting point processes are widely used to model discrete event data in which past and current events affect the likelihood of future events. In this study, we detect abrupt changes in the coefficient matrices of discrete-time high-dimensional self-exciting Poisson processes, which have yet to be studied because of the theoretical and computational challenges in the nonstationary and high-dimensional nature of the underlying process. We propose a penalized dynamic programming approach, supported by a theoretical rate analysis and numerical evidence.

This article examines the properties of the variance risk premium (VRP). We propose a flexible asset pricing model that captures co-jumps in prices and volatility, and self-exciting jump clustering. We estimate the model on equity returns and variance swap rates at different horizons. The total VRP is negative and has a downward-sloping term structure, while its jump component displays an upward-sloping term structure. The abrupt and persistent response of the short-term jump VRP to extreme events makes this specific premium a proxy for investors' fear of a market crash. Furthermore, the use of the VRP level and slope, and of its components, helps improve the short-run predictability of equity excess returns.

Information diffusion occurs on microblogging platforms like Twitter as retweet cascades. When a tweet is posted, it may be retweeted and hence-forth further retweeted, and the retweeting process continues iteratively and indefinitely. A natural measure of the popularity of a tweet is the number of retweets it generates. Accurate predictions of tweet popularity can assist Twitter to rank contents more effectively and facilitate the assessment of potential for marketing and campaigning strategies. In this paper, we propose a model called the Marked Self-Exciting Process with Time-Dependent Excitation Function, or MaSEPTiDE for short, to model the retweeting dynamics and to predict the tweet popularity. Our model does not require expensive feature engineering but is capable of leveraging the observed dynamics to accurately predict the future evolution of retweet cascades. We apply our proposed methodology on a large amount of Twitter data and report substantial improvement in prediction performance over existing approaches in the literature.

We develop a stochastic volatility framework for modeling multiple currencies based on CBI-time-changed Lévy processes. The proposed framework captures the typical risk characteristics of FX markets and is coherent with the symmetries of FX rates. Moreover, due to the self-exciting behavior of CBI processes, the volatilities of FX rates exhibit self-exciting dynamics. By relying on the theory of affine processes, we show that our approach is analytically tractable and that the model structure is invariant under a suitable class of risk-neutral measures. A semi-closed pricing formula for currency options is obtained by Fourier methods. We propose two calibration methods, also by relying on deep-learning techniques, and show that a simple specification of the model can achieve a good fit to market data on a currency triangle.