Abstract

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.