Thanks to the ever-advancing information-processing horsepower, a broad spectrum of large-scale and high-frequency data with complex spatiotemporal dependent structures have become available. Stochastic processes serve as essential tools for modeling such data. Importantly, most of them involve non-Gaussian characters often caused by jumps with various types of intensities and/or state-dependent structures. To create and deal with such models in an efficient manner, it is urgent and imperative to develop statistical methodologies based on both advanced mathematical statistics and stochastic analyses for stochastic processes. Simple combinations and/or adaptations of the currently available models and statistics are not enough, and associated bodies of theories are yet to be developed.

Our session is intended to present the state-of-the-art of this quite active area of research. Specifically, our objectives include general Lévy-driven stochastic processes and various kinds of models driven by a Hawkes-type self-exciting point process, featuring how to statistically manage various kinds of non-Gaussianity to be able to deal with the above-mentioned dependent data in theoretically rigorous yet practical and transparent manners. Our session consists of four specialists.

Arnaud Gloter from France (Université d'Évry Val d'Essonne) is an expert in statistics for stochastic processes with jumps having complicated dependence and self-exciting structure. The topic has recently been highlighted as a new direction of research on financial large-scale data, neurosciences, and earthquake-prediction analysis, to mention just a few. His recent contributions include several contemporary cornerstones not only in mathematical statistics but also in stochastic calculus of jump processes.

Hiroki Masuda from Japan (The University of Tokyo) specializes in statistics for Lévy-driven models. His recent research objectives include quasi-likelihood analyses of Gaussian and non-Gaussian types and their complementary properties and robustification, regularized estimation of both sparse and non-sparse types, and information criteria. He is primarily interested in constructing statistical methods that well balance theory and computation in practice through creating easy-to-use statistics.

Lorenzo Mercuri from Italy (University of Milan) is an active researcher with expertise in not only statistical modeling for a general class of point-process regressions but also in building efficient numerical algorithms and software development. Recently, he is working on several CARMA-type models, along with developing some tractable multivariate non-Gaussian distributions, partly called mixed tempered stable. Further, he is a core developer of the YUIMA R package designed for simulation and statistics of various kinds of stochastic processes.

Ioane Muni Toke from France (CentraleSupélec, chair of Quantitative Finance) is an expert in quantitative finance. He has proposed many statistical methods for modeling order flows in limit order books based on marked point process type and developed related quasi-likelihood analysis. Through extensive real data analyses, he recently proved some causality phenomena when one applies Hawkes type process to financial market data and clarified the importance of statistical model assessment.

With the above-mentioned backgrounds, all the talks in our session provide us with advanced contemporary machinery from not only theoretical but also computational points of view. They are closely connected in the field of mathematical statistics for stochastic processes, and a large amount of synergistic development can be expected through information sharing. They should serve as solid theoretical and computational foundations for future developments in related directions.

Arnaud Gloter will present nonparametric inference of the coefficients of a self-exciting jump-diffusion process based on high-frequency observations over a long-time horizon. The proposed inference procedure consists of three steps: first, the volatility coefficient is estimated over an appropriate linear subspace through a smoothed truncation methodology. A theoretical bound for the empirical risk is obtained, making a regularity-based adaptive estimation possible subsequently. Second, an estimator of a sum between the volatility and the coefficient of self-exciting jumps is obtained through the conditional expectation of the jump intensity. Therein, an oracle inequality for an adaptive estimator is obtained. Finally, a methodology is given to recover the jump-component function. Extensive simulation studies are conducted to measure the accuracy of the proposed estimators in practice.

Hiroki Masuda will talk about the noise-inference problem for non-Gaussian Lévy-driven stochastic differential equation (SDE). By making extensive use of appropriate quasi-likelihood functions, a stochastic expansion for the unit-time residuals functionals is derived. The results reveal the quantitative effect of plugging in the estimators of the coefficients, paving a flexible way toward practical and theoretically validated inference procedures for the driving-noise characteristics. In addition, we will also present a combined usage of the different quasi-likelihoods for asymptotically efficient inference with bypassing heavy numerical optimization and show how one can robustify the quasi-likelihood against jumps and/or outliers in a simple manner. Further, we demonstrate the implementation of the theoretical results on computers.

Lorenzo Mercuri will present the novel use of CARMA(p,q) models for the intensity of point processes. Point processes are vital to describe the dynamics of observed event times. A standard Hawkes model with an exponential kernel entails a strictly decreasing behavior of the autocorrelation function. However, the monotonicity assumption on the autocorrelation function is often empirically rejected. To reproduce more realistic dependence structures, he will introduce a generalization of the Hawkes process, called CARMA(p,q)-Hawkes process, by incorporating a CARMA(p,q) model for the intensity. In that framework, an estimation procedure based on the distance between theoretical and empirical autocorrelation functions is given together with its asymptotic properties.

Ioane Muni Toke will talk about Markov-switching Hawkes processes with applications to modeling financial microstructure. He built a multidimensional point process with Hawkes intensities through a hidden Markov chain. Estimation of both process intensities and state transition probabilities is carried out using an EM algorithm, and an alternative Bayesian approach is also provided. In-depth numerical and computational analyses of the performances of the estimation algorithms are presented. The results extend and improve previous modeling propositions developed in more restricted frameworks, for example with restricted excitation kernels. The model is applied to analyze equity and cryptocurrency market data, and it turns out to be an efficient tool to retrieve financially interpretable market states described by different intensity regimes.

Organiser: Prof. Hiroki Masuda 

Chair: Dr Kengo Kamatani  

Speaker: Lorenzo Mercuri 

Speaker: Prof. Arnaud Gloter 

Speaker: Ioane Muni Toke  

Speaker: Prof. Hiroki Masuda