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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.

In this paper we focus on a class of linear Hawkes processes with general immigrants. These are counting processes with shot-noise intensity, including self-excited and externally excited patterns. For such processes, we introduce the concept of the age pyramid which evolves according to immigration and births. The virtue of this approach that combines an intensity process definition and a branching representation is that the population age pyramid keeps track of all past events. This is used to compute new distribution properties for a class of Hawkes processes with general immigrants which generalize the popular exponential fertility function. The pathwise construction of the Hawkes process and its underlying population is also given.

The aim of this thesis is to address the real need of predicting the prices of stocks. In fact, the randomness governing the evolution of prices is, for financial players like market makers, one of the largest sources of risk. In this context, we highlight the possibility of reducing the uncertainty of the future prices using appropriate mathematical models. This study was made possible by a large base of high frequency data and a powerful computational grid provided by the Automatic Market Making team at BNP Paribas. In this paper, we present only the results of high frequency tests. Tests are of less scientific interest in the academic world and are confidential. Therefore, these results will be deliberately omitted.In the first chapter, the background and the objectives of this study are presented along with the different methods used and the main results obtained.The focus of chapter 2 is on the contribution of technological superiority in high frequency trading. In order to do this, an omniscient trader is simulated and the total gain over three years is calculated. The obtained gain is very modest and reflects the limited contribution of technology in high frequency trading. This result underlines the primary role of research and modeling in this field.In Chapter 3, the predictability of prices using some order book indicators is studied. Using conditional expectations, the empirical evidence of the statistical dependencies between the prices and indicators is presented. The importance of these dependencies results from the simplicity of the method, eliminating any risk of over fitting the data. Then the combination of the various indicators is tested using a linear regression and the various numerical and statistical problems associated with this method are analyzed. Finally, it can be concluded that the prices are predictable for a period of a few minutes and the assumption of market efficiency is questioned.In Chapter 4, the mechanism of price formation from the arrival of events in the order book is investigated. The orders are classified in twelve types and their statistical properties are analyzed. The dependencies between these different types of orders are studied and a model of order book in line with the empirical observations is proposed. Finally, this model is used to predict prices and confirm the assumption of market inefficiency suggested in Chapter 3., Le but de cette thèse est de répondre au vrai besoin de prédire les fluctuations futures des prix d'actions. En effet, l'aléatoire régissant ces fluctuations constitue pour des acteurs de la finance, tels que les Market Maker, une des plus grandes sources de risque. Tout au long de cette étude, nous mettons en évidence la possibilité de réduire l'incertitude sur les prix futurs par l'usage des modèles mathématiques appropriés. Cette étude est rendue possible grâce à une grande base de données financières et une puissante grille de calcul mises à notre disposition par l'équipe Automatic Market Making de BNP Paribas. Dans ce document, nous présentons uniquement les résultats de la recherche concernant le trading haute fréquence. Les résultats concernant la partie basse fréquence présentent un intérêt scientifique moindre pour le monde académique et rentrent par ailleurs dans le cadre des résultats confidentiels. Ces résultats seront donc volontairement omis.Dans le premier chapitre, nous présentons le contexte et les objectifs de cette étude. Nous présentons, également, les différentes méthodes utilisées, ainsi que les principaux résultats obtenus. Dans le chapitre 2, nous nous intéressons à l'apport de la supériorité technologique en trading haute fréquence. Dans ce but, nous simulons un trader ultra rapide, omniscient, et agressif, puis nous calculons son gain total sur 3 ans. Les gains obtenus sont très modestes et reflètent l'apport limité de la technologie en trading haute fréquence. Ce résultat souligne l'intérêt primordial de la recherche et de la modélisation dans ce domaine.Dans le chapitre 3, nous étudions la prédictibilité des prix à partir des indicateurs de carnet d'ordre. Nous présentons, à l'aide des espérances conditionnelles, des preuves empiriques de dépendances statistiques entre les prix et les différents indicateurs. L'importance de ces dépendances résulte de la simplicité de la méthode, éliminant tout risque de surapprentissage des données. Nous nous intéressons, ensuite, à la combinaison des différents indicateurs par une régression linéaire et nous analysons les différents problèmes numériques et statistiques liés à cette méthode. Enfin, nous concluons que les prix sont prédictibles pour un horizon de quelques minutes et nous mettons en question l'hypothèse de l'efficience du marché.Dans le chapitre 4, nous nous intéressons au mécanisme de formation du prix à partir des arrivés des évènements dans le carnet d'ordre. Nous classifions les ordres en douze types dont nous analysons les propriétés statistiques. Nous étudions par la suite les dépendances entre ces différents types d'ordres et nous proposons un modèle de carnet d'ordre en ligne avec les observations empiriques. Enfin, nous utilisons ce modèle pour prédire les prix et nous appuyons l'hypothèse de la non-efficience des marchés, suggérée au chapitre 3.

The aim of this thesis is to address the real need of predicting the prices of stocks. In fact, the randomness governing the evolution of prices is, for financial players like market makers, one of the largest sources of risk. In this context, we highlight the possibility of reducing the uncertainty of the future prices using appropriate mathematical models. This study was made possible by a large base of high frequency data and a powerful computational grid provided by the Automatic Market Making team at BNP Paribas. In this paper, we present only the results of high frequency tests. Tests are of less scientific interest in the academic world and are confidential. Therefore, these results will be deliberately omitted.In the first chapter, the background and the objectives of this study are presented along with the different methods used and the main results obtained.The focus of chapter 2 is on the contribution of technological superiority in high frequency trading. In order to do this, an omniscient trader is simulated and the total gain over three years is calculated. The obtained gain is very modest and reflects the limited contribution of technology in high frequency trading. This result underlines the primary role of research and modeling in this field.In Chapter 3, the predictability of prices using some order book indicators is studied. Using conditional expectations, the empirical evidence of the statistical dependencies between the prices and indicators is presented. The importance of these dependencies results from the simplicity of the method, eliminating any risk of over fitting the data. Then the combination of the various indicators is tested using a linear regression and the various numerical and statistical problems associated with this method are analyzed. Finally, it can be concluded that the prices are predictable for a period of a few minutes and the assumption of market efficiency is questioned.In Chapter 4, the mechanism of price formation from the arrival of events in the order book is investigated. The orders are classified in twelve types and their statistical properties are analyzed. The dependencies between these different types of orders are studied and a model of order book in line with the empirical observations is proposed. Finally, this model is used to predict prices and confirm the assumption of market inefficiency suggested in Chapter 3., Le but de cette thèse est de répondre au vrai besoin de prédire les fluctuations futures des prix d'actions. En effet, l'aléatoire régissant ces fluctuations constitue pour des acteurs de la finance, tels que les Market Maker, une des plus grandes sources de risque. Tout au long de cette étude, nous mettons en évidence la possibilité de réduire l'incertitude sur les prix futurs par l'usage des modèles mathématiques appropriés. Cette étude est rendue possible grâce à une grande base de données financières et une puissante grille de calcul mises à notre disposition par l'équipe Automatic Market Making de BNP Paribas. Dans ce document, nous présentons uniquement les résultats de la recherche concernant le trading haute fréquence. Les résultats concernant la partie basse fréquence présentent un intérêt scientifique moindre pour le monde académique et rentrent par ailleurs dans le cadre des résultats confidentiels. Ces résultats seront donc volontairement omis.Dans le premier chapitre, nous présentons le contexte et les objectifs de cette étude. Nous présentons, également, les différentes méthodes utilisées, ainsi que les principaux résultats obtenus. Dans le chapitre 2, nous nous intéressons à l'apport de la supériorité technologique en trading haute fréquence. Dans ce but, nous simulons un trader ultra rapide, omniscient, et agressif, puis nous calculons son gain total sur 3 ans. Les gains obtenus sont très modestes et reflètent l'apport limité de la technologie en trading haute fréquence. Ce résultat souligne l'intérêt primordial de la recherche et de la modélisation dans ce domaine.Dans le chapitre 3, nous étudions la prédictibilité des prix à partir des indicateurs de carnet d'ordre. Nous présentons, à l'aide des espérances conditionnelles, des preuves empiriques de dépendances statistiques entre les prix et les différents indicateurs. L'importance de ces dépendances résulte de la simplicité de la méthode, éliminant tout risque de surapprentissage des données. Nous nous intéressons, ensuite, à la combinaison des différents indicateurs par une régression linéaire et nous analysons les différents problèmes numériques et statistiques liés à cette méthode. Enfin, nous concluons que les prix sont prédictibles pour un horizon de quelques minutes et nous mettons en question l'hypothèse de l'efficience du marché.Dans le chapitre 4, nous nous intéressons au mécanisme de formation du prix à partir des arrivés des évènements dans le carnet d'ordre. Nous classifions les ordres en douze types dont nous analysons les propriétés statistiques. Nous étudions par la suite les dépendances entre ces différents types d'ordres et nous proposons un modèle de carnet d'ordre en ligne avec les observations empiriques. Enfin, nous utilisons ce modèle pour prédire les prix et nous appuyons l'hypothèse de la non-efficience des marchés, suggérée au chapitre 3.

Modeling and forecasting realized volatility is of paramount importance. Previous studies have examined the role of both the continuous and jump components of volatility in forecasting. This paper considers how to use index level jumps and cojumps across index constituents for forecasting index level volatility. In combination with the magnitude of past index jumps, the intensity of both index jumps and cojumps are examined. Estimated jump intensity from a point process model is used within a forecasting regression framework. Even in the presence of the diffusive part of total volatility, and past jump size, intensity of both index and cojumps are found to significantly improve forecast accuracy. An important contribution is that information relating to the behaviour of underlying constituent stocks is useful for forecasting index level behaviour. Improvements in forecast performance are particularly apparent on the days when jumps or cojumps occur, or when markets are turbulent.

This paper investigates the dynamic behaviour of jumps in financial prices and volatility. The proposed model is based on a standard jump diffusion process for price and volatility augmented by a bivariate Hawkes process for the two jump components. The latter process speci.es a joint dynamic structure for the price and volatility jump intensities, with the intensity of a volatility jump also directly affected by a jump in the price. The impact of certain aspects of the model on the higher-order conditional moments for returns is investigated. In particular, the differential effects of the jump intensities and the random process for latent volatility itself, are measured and documented. A state space representation of the model is constructed using both financial returns and non-parametric measures of integrated volatility and price jumps as the observable quantities. Bayesian inference, based on a Markov chain Monte Carlo algorithm, is used to obtain a posterior distribution for the relevant model parameters and latent variables, and to analyze various hypotheses about the dynamics in, and the relationship between, the jump intensities. An extensive empirical investigation using data based on the S&P500 market index over a period ending in early-2013 is conducted. Substantial empirical support for dynamic jump intensities is documented, with predictive accuracy enhanced by the inclusion of this type of specification. In addition, movements in the intensity parameter for volatility jumps are found to track key market events closely over this period.

We propose a modeling framework which allows for creating probability predictions on a future market crash in the medium term, like sometime in the next five days. Our framework draws upon noticeable similarities between stock returns around a financial market crash and seismic activity around earthquakes. Our model is incorporated in an Early Warning System for future crash days. Testing our EWS on S&P 500 data during the recent financial crisis, we find positive Hanssen-Kuiper Skill Scores. Furthermore our modeling framework is capable of exploiting information in the returns series not captured by well known and commonly used volatility models. EWS based on our models outperform EWS based on the volatility models forecasting extreme price movements, while forecasting is much less time-consuming.

While the presence of clustering in crime and security event data is well established, the mechanism(s) by which clustering arises is not fully understood. Both contagion models and history independent correlation models are applied, but not simultaneously. In an attempt to disentangle contagion from other types of correlation, we consider a Hawkes process with background rate driven by a log Gaussian Cox process. Our inference methodology is an efficient Metropolis adjusted Langevin algorithm for filtering of the intensity and estimation of the model parameters. We apply the methodology to property and violent crime data from Chicago, terrorist attack data from Northern Ireland and Israel, and civilian casualty data from Iraq. For each data set we quantify the uncertainty in the levels of contagion vs. history independent correlation., Comment: Published in at http://dx.doi.org/10.1214/13-AOAS647 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We introduce a stochastic volatility model with self-exciting jump intensity to capture the change in pricing dynamic triggered by big negative stock returns. The stochastic variance and jump intensity, and their risk premium are estimated jointly from daily stock returns and option data over 2007-2010. The model is calibrated to cumulants implied from option prices instead of option prices directly. We find evidence that the time varying jump intensity plays a very important role in the sub-prime crisis and explained most of the risk premium, while in other calmer periods, stochastic variance accounts for most of the risk premium.

In this paper, we propose a stochastic process, which is a Cox-Ingersoll-Ross process with Hawkes jumps. It can be seen as a generalization of the classical Cox-Ingersoll-Ross process and the classical Hawkes process with exponential exciting function. Our model is a special case of the affine point processes. Laplace transforms and limit theorems have been obtained, including law of large numbers, central limit theorems and large deviations., Comment: 14 pages

International audience; In this work, we adopt Spread constrained Limit Order Book Hawkes Process (ScLOBHP) introduced by Zheng, Roueff and Abergel (2012) to model the financial high frequency data. The best limit prices in a limit order book are modelled using ScLOBHP since the spread plays a constraint in the transition probability of itself. For large stocks, we show that most of the best limit price changes is only change of one tick. We propose a basic ScLOBHP considering only the best limit prices' one tick change events and calibrate the parameters by maximum likelihood estimation. The cross-excitation relationship between different events are proven to be stable over a long period. However, the immigrant intensity and the exponential decaying fertility rate are likely to be determined by the liquidity. Finally, we propose a more sophisticated ScLOBHP to model extreme events such as multi-ticks price changes in order to provide a view on the impact of extreme events. We highlight that two-ticks price change events have an impact on one-tick price change events, nevertheless, few impact is detected in reverse.

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., 13 pages

A predictive model of terrorist activity is developed by examining the daily number of terrorist attacks in Indonesia from 1994 through 2007. The dynamic model employs a shot noise process to explain the self-exciting nature of the terrorist activities. This estimates the probability of future attacks as a function of the times since the past attacks. In addition, the excess of nonattack days coupled with the presence of multiple coordinated attacks on the same day compelled the use of hurdle models to jointly model the probability of an attack day and corresponding number of attacks. A power law distribution with a shot noise driven parameter best modeled the number of attacks on an attack day. Interpretation of the model parameters is discussed and predictive performance of the models is evaluated., Comment: Published in at http://dx.doi.org/10.1214/11-AOAS513 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In an earlier study (Van Lieshout and Stein, in press) we postulated the existence of two major earthquakes in Kashmir instead of a single one, based upon the pattern of aftershocks. In this note we explore this pattern further by fitting several spatial point pattern models. In particular we discuss the Hawkes and the trigger process models for earthquake aftershock sequences following the Kashmir catastrophe in 2005. The minimum contrast method is used for estimation of the parameters. The study shows that the trigger model fits better than the Hawkes model. The most likely number of main shocks is rounded to 2 generating the almost 200 aftershocks, whereas the Hawkes model would estimate a parent process of approximately 18 parents with on average 10 descendants. We conclude that the spatial pattern of aftershocks can best be understood as being generated by the trigger point process model.

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.

The aim of this paper is to provide a new method for the detection of either favored or avoided distances between genomic events along DNA sequences. These events are modeled by a Hawkes process. The biological problem is actually complex enough to need a nonasymptotic penalized model selection approach. We provide a theoretical penalty that satisfies an oracle inequality even for quite complex families of models. The consecutive theoretical estimator is shown to be adaptive minimax for H\"{o}lderian functions with regularity in $(1/2,1]$: those aspects have not yet been studied for the Hawkes' process. Moreover, we introduce an efficient strategy, named Islands, which is not classically used in model selection, but that happens to be particularly relevant to the biological question we want to answer. Since a multiplicative constant in the theoretical penalty is not computable in practice, we provide extensive simulations to find a data-driven calibration of this constant. The results obtained on real genomic data are coherent with biological knowledge and eventually refine them., Comment: Published in at http://dx.doi.org/10.1214/10-AOS806 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

This article concerns a perfect simulation algorithm for unmarked and marked Hawkes processes. The usual stratihtforward simulation algorithm suffers from edge effects, whereas our perfect simulation algorithm does not. By viewing Hawkes processes as Poisson cluster processes and using their branching and conditional independence structure, useful approximations of the distribution function for the length of a cluster are derived. This is used to construct upper and lower processes for the perfect simulation algorithm. Examples of applications and empirical results are presented.

When dealing with classical spike train analysis, the practitioner often performs goodness-of-fit tests to test whether the observed process is a Poisson process, for instance, or if it obeys another type of probabilistic model (Yana et al. in Biophys. J. 46(3):323–330, 1984; Brown et al. in Neural Comput. 14(2):325–346, 2002; Pouzat and Chaffiol in Technical report, http://arxiv.org/abs/arXiv:0909.2785 , 2009). In doing so, there is a fundamental plug-in step, where the parameters of the supposed underlying model are estimated. The aim of this article is to show that plug-in has sometimes very undesirable effects. We propose a new method based on subsampling to deal with those plug-in issues in the case of the Kolmogorov–Smirnov test of uniformity. The method relies on the plug-in of good estimates of the underlying model that have to be consistent with a controlled rate of convergence. Some nonparametric estimates satisfying those constraints in the Poisson or in the Hawkes framework are highlighted. Moreover, they share adaptive properties that are useful from a practical point of view. We show the performance of those methods on simulated data. We also provide a complete analysis with these tools on single unit activity recorded on a monkey during a sensory-motor task. Electronic Supplementary Material The online version of this article (doi:10.1186/2190-8567-4-3) contains supplementary material.

We construct a ‘reflexivity’ index to measure the activity generated endogenously within a market for cryptocurrencies. For this purpose, we fit a univariate self-exciting Hawkes process with two classes of parametric kernels to high-frequency trading data. A parsimonious model of both endogenous and exogenous dynamics enables a direct comparison with exchanges for traditional asset classes, in terms of identified branching ratios. We also formulate a ‘Hawkes disorder problem,’ as a generalization of the established Poisson disorder problem, and provide a simulation-based approach to determining an optimal observation horizon. Our analysis suggests that Bitcoin mid-price dynamics feature long-memory properties, well explained by the power-law kernel, at a level of criticality similar to the fiat-currency market.

In this article, we review known results and present new ones concerning the power spectra of large classes of signals and random fields driven by an underlying point process, such as spatial shot noises (with random impulse response and arbitrary basic stationary point processes described by their Bartlett spectra) and signals or fields sampled at random times or points (where the sampling point process is again quite general). We also obtain the Bartlett spectrum for the general linear Hawkes spatial branching point process (with random fertility rate and general immigrant process described by its Bartlett spectrum). We then obtain the Bochner spectra of general spatial linear birth and death processes. Finally, we address the issues of random sampling and linear reconstruction of a signal from its random samples, reviewing and extending former results.