Some limit theorems for Hawkes processes and application to financial statistics

The occurrence patterns of important words found in six texts (one historical pamphlet and five renowned academic books) are analyzed using both univariate and multivariate Hawkes processes. By treating the occurrence patterns as binary time-series data along the texts, we investigate how effectively univariate and multivariate Hawkes processes capture the characteristics of these word occurrence signals. Through maximum likelihood estimation and subsequent simulations, we found that the multivariate Hawkes process clearly outperforms the univariate Hawkes process in modeling word occurrence signals. Moreover, we found that the multivariate Hawkes process can provide a Hawkes graph, which serves as an intuitive representation of the relationships between concepts appearing in the analyzed text. Furthermore, our study demonstrates that the importance of concepts within a given text can be quantitatively estimated based on the optimized parameter values of the multivariate Hawkes process.

As an extension of self-exciting Hawkes process, the multivariate Hawkes process models counting processes of different types of random events with mutual excitement. In this paper, we present a perfect sampling algorithm that can generate i.i.d. stationary sample paths of multivariate Hawkes process without any transient bias. In addition, we provide an explicit expression of algorithm complexity in model and algorithm parameters and provide numerical schemes to find the optimal parameter set that minimizes the complexity of the perfect sampling algorithm.

How Can Our Tweets Go Viral? Point-Process Modelling of Brand Content

This work contributes to the theory and applications of Hawkes processes. We introduce and examine a new class of Hawkes processes that we call generalized Hawkes processes, and their special subclass -- the generalized multivariate Hawkes processes (GMHPs). GMHPs are multivariate marked point processes that add an important feature to the family of the (classical) multivariate Hawkes processes: they allow for explicit modelling of simultaneous occurrence of excitation events coming from different sources, i.e. caused by different coordinates of the multivariate process. We study the issue of existence of a generalized Hawkes process, and we provide a construction of a specific generalized multivariate Hawkes process. We investigate Markovian aspects of GMHPs, and we indicate some plausible important applications of GMHPs.

A Point Process Framework for Predicting Popularity of Content in Online Social Networks

Hawkes processes are popular for modeling correlated temporal sequences that exhibit mutual-excitation properties. Existing approaches such as feature-enriched processes or variations of Multivariate Hawkes processes either fail to describe the exact mutual influence between sequences or become computational inhibitive in most real-world applications involving large dimensions. Incorporating additional geometric structure in the form of graphs into Hawkes processes is an effective and efficient way for improving model prediction accuracy. In this paper, we propose the Geometric Hawkes Process (GHP) model to better correlate individual processes, by integrating Hawkes processes and a graph convolutional recurrent neural network. The deep network structure is computational efficient since it requires constant parameters that are independent of the graph size. The experiment results on real-world data show that our framework outperforms recent state-of-art methods.

This study aims to understand the contagion effect of the major equity market sentiment events, defined as jumps in the VIX index, on cryptocurrency price jumps and the corresponding feedback effect on investors' sentiment. Using recent high frequency intraday data with multivariate Hawkes processes, we find that several noteworthy contagion effects exist between Bitcoin and the market sentiment proxied by the VIX index. First of all, we find that positive market sentiment jumps tend to trigger a moderate level of cross-contagion on both positive and negative jumps in the Bitcoin market, whereas negative market sentiment jumps have no contagion effect on the Bitcoin prices. We also find that the Bitcoin market shows stronger positive self-contagion and cross-contagion effects than the equity market, in general. We also observe a lasting fear of missing out (FOMO) phenomenon in Bitcoin. It is supported by the evidence that the effect of positive jumps in the Bitcoin market lasts three times as long as that in both the equity market and the negative Bitcoin price jumps, and these positive jumps in Bitcoin may serve as a harbinger of equity market movement. These findings provide a better understanding of risk control and policy guidance for the cryptocurrency market.

We propose a method for the modelling of the propagation of text data in web site space among some groups by using a multivariate Hawkes process with a sparse structure. For estimation, we introduced a hybrid method using a quasi-maximum likelihood estimator (QMLE) and a $L^1$-penalized QMLE. As a real example, we investigated posts on a Japanese web service about uncomfortable gender experiences, which we classified into 12 groups by age and sex, and we calculated the magnitude of the correlation between each group. In addition, we visualized the propagation structure of posts in each group by summarizing the results in a directed Hawkes graph and a heat map of time integrals of kernel functions.

In this paper, we focus on a new generalization of multivariate general compound Hawkes process (MGCHP), which we referred to as the multivariate general compound point process (MGCPP). Namely, we applied a multivariate point process to model the order flow instead of the Hawkes process. The law of large numbers (LLN) and two functional central limit theorems (FCLTs) for the MGCPP were proved in this work. Applications of the MGCPP in the limit order market were also considered. We provided numerical simulations and comparisons for the MGCPP and MGCHP by applying Google, Apple, Microsoft, Amazon, and Intel trading data.

If a known transformation of a random variable X is normally distributed with mean {mu} and variance {sigma}{sup 2}, then the mean, variance, and any other distributional property of X can be expressed in terms of {mu} and {sigma}{sup 2}. For example, if X is lognormally distributed, i.e., X {approx} {Lambda}({mu}, {sigma}{sup 2}) or (equivalently) Y = log(X) {approx} N({mu}, {sigma}{sup 2}), then the expected value, variance, median, and mode of X are, respectively, E(X) = exp({mu} + {sigma}{sup 2}/2), var (X) = exp(2{mu} + {sigma}{sup 2})(exp({sigma}{sup 2}) - 1), med(X) = exp({mu}), and mode(X) = exp({mu} - {sigma}{sup 2}). Exact and optimal (uniformly most accurate unbiased) confidence limit procedures have been developed for linear functions of {mu} and {sigma}{sup 2} (Land, 1971, 1973) and, therefore, because confidence limits for a parameter are invariant under smooth, monotone transformations of that parameter, for the mean and mode of a lognormal distribution. In fact, the lognormal distribution is the only one whose mean can be expressed as a function of a non-trivial linear combination of {mu} and {sigma}{sup 2} (Land, 1971), but other functions, including those arising in connection with other normalizing transformations, can be approximated locally by linear functions for which exact limits can be constructed that define approximate limits for the original parametric functions of interest (Land, 1974, 1988). Tables have been published to facilitate the calculation of confidence limits for arbitrary linear functions of {mu} and {sigma}{sup 2} (Land, 1975), but their use is often tedious, requiring repeated interpolation and calculation. An unpublished Fortran program to compute confidence limits directly from sample estimates of p and u2 has been available from the second author, and has been used by a number of investigators to analyze lognormal data sets. The present paper introduces a more efficient computational algorithm, and documents the program for prospective users. An option has been added which makes it easy for the user to generate tables of confidence limits. Finally, the accuracy of the program has been thoroughly evaluated in terms of coverage probabilities for a wide range of parameter values.

Next-generation wireless networks have been designed to transport integrated multimedia services based on a cellular extension of a packet-switched architecture using variants of the Internet protocol (IP). Each call, arriving to or active within the network, carries demand for one or more services in parallel, where each service type has a guaranteed quality of service (QoS). Admission of new calls to the wireless IP network (WIN) from the gateway of a wired network or from a mobile subscriber (MS) is allowed by call admission control (CAC) procedures. MS roaming among the nodes of the WIN is controlled by handoff procedures between base stations (BSs), or BS controllers (BSCs), and the MSs. Both handoff and CAC procedures are typically embedded in the media access control (MAC) and logical link control (LLC) layers of the WIN protocol stack. Performance metrics, such as, probabilities of call blocking and handoff, control procedures within the WIN protocol. Earlier published results by the author present a method for the performance analysis of wireless multimedia networks based on multivariate point-process (MVPP) models and their semi-martingale representations. The models describe the finite-horizon, transient behavior of the packet flows in integrated multimedia traffic. This paper extends the earlier work to focus on the optimization of real-time control procedures embedded in the WIN protocol stack. The point processes corresponding to the packet flow of each service type and to collateral random network events are decomposed into right-continuous, pure jump processes and predictable, integrated random rate processes. Control of the set of point processes describing WIN behavior is implemented through the construction of an absolutely continuous change of a reference probability measure on network events to a controlled measure. With respect to the constructed controlled measure, the integrated random rates in the semi-martingale representations of the network point processes explicitly depend on the parameters of the protocol mechanisms for packet access, routing, switching, handoff, power control, and other resource allocations. The MVPP control approach via constructed probability measure also supports predictive models of packet flows that incorporate measurement-based estimates of the probability distributions for voice, video, data, and other Internet traffic as well as radio path distortions. Optimization of the performance metrics, represented in terms of the MVPPs and the control parameters of candidate WIN protocols for multimedia services, are discussed for both partial and complete observations of network events. The models are further used to develop optimal, recursive stochastic filters of network state, based on partial or incomplete observations of packet flow dynamics.

Modelling Financial High Frequency Data Using Point Processes

In this paper, we present a framework for fitting multivariate Hawkes processes for large-scale problems, both in the number of events in the observed history n and the number of event types d (i.e. dimensions). The proposed Scalable Low-Rank Hawkes Process (SLRHP) framework introduces a low-rank approximation of the kernel matrix that allows to perform the nonparametric learning of the d2 triggering kernels in at most O(ndr2) operations, where r is the rank of the approximation (r ≪ d, n). This comes as a major improvement to the existing state-of-the-art inference algorithms that require O(nd2) operations. Furthermore, the low-rank approximation allows SLRHP to learn representative patterns of interaction between event types, which is usually valuable for the analysis of complex processes in real-world networks.

Affine Point Processes and Portfolio Credit Risk

This thesis focuses on the development and analysis of general methods for the design of optimal discrete PID control strategies for multivariable glass furnace processes, where standard genetic algorithms (SGAs) are applied to optimise specially formulated objective functions. Furthermore, a strong emphasis is given on the realistic model parameters identi cation method, which is illustrated to be applicable to a wide range of higher order model parameters identi cation problems. A complete, realistic and continuous excess oxygen model with nonlinearity effect was developed and the model parameters were identified. The developed excess oxygen model consisted of three sub-models to characterise the real plant response. The developed excess oxygen model was evaluated and compared with real plant dynamic response data, which illustrated the high degree of accuracy of the developed model. A new technique named predetermined time constant approximation was proposed to make an assumption on the initial value of a predetermined time constant, whose motive is to facilitate the SGAs to explore and exploit an optimal value for higher order of continuous model's parameters identi cation. Also, the proposed predetermined time constant approximation technique demonstrated that the population diversity is well sustained while exploring the feasible search region and exploiting to an optimal value. In general, the proposed method improves the SGAs convergence rate towards the global optimum and illustrated the effectiveness. An automatic tuning of decentralised discrete PID controllers for multivariable processes, based on SGAs, was proposed. The main improvement of the proposed technique is the ability to enhance the control robustness and to optimise discrete PID parameters by compensating the loop interaction of a multivariable process. This is attained by adding the individually optimised objective function of glass temperature and excess oxygen processes as one objective function, to include the total effect of the loop interaction by applying step inputs on both set points, temperature and excess oxygen, at two different time periods in one simulation. The effectiveness of the proposed tuning technique was supported by a number of simulation results using two other SGAs conventional tuning techniques with 1st and 2nd order control oriented models. It was illustrated that, in all cases, the resulting discrete PID control parameters completely satisfied all performance specifications. A new technique to minimise the fuel consumption for glass furnace processes while sustaining the glass temperature is proposed. This proposed technique is achieved by reducing the excess oxygen within the optimum thermal efficiency region within 1.7% to 3.2%, which is approximately equal to about 10% to 20% of excess air. Therefore, by reducing the excess oxygen set point within the optimum region, 2.45% to 2%, the fuel consumption is minimised from 0:002942kg/sec to 0:002868kg/sec while the thermal efficiency of the glass temperature is sustained at the desired set point (1550K). In addition, a reduction in excess oxygen within methane combustion guidelines will assure that undesirable emissions are in control throughout the combustion process. The efficiencies of the proposed technique were supported by a number of simulation results applying the three SGAs controller tuning techniques. It was illustrated that, in all cases, the fraction of excess oxygen reduction results in a great minimisation of fuel consumption over long plant operating periods.