A0870
Title: Invariant density estimation of self-exciting jump-diffusion
Authors: Chiara Amorino - University Pompeu Fabra (Spain) [presenting]
Arnaud Gloter - Universite d Evry Val d Essonne (France)
Charlotte Dion - Sorbonne Universite (France)
Sarah Sarah Lemler - Ecole CentraleSupelec (France)
Abstract: Results are presented on estimating the invariant density associated with $(X_t, \lambda_t)_{t \ge 0}$, where $X$ is diffusion with jumps driven by a multivariate nonlinear Hawkes process, and $\lambda$ is a piecewise deterministic Markov process (PDMP) defining the stochastic intensity. Estimating the invariant density is crucial due to its applications in physics and numerical methods, particularly Markov chain Monte Carlo. Non-parametric estimation for the stationary measure of a continuous mixing process is a well-established yet evolving topic. Estimating the invariant density $\pi(x,y)$ of $(X, \lambda)$ is proposed using kernel density estimation, assuming a continuous record of $X$ is available. Accuracy is measured by the pointwise $L^2$ error, requiring pre-estimation of $\lambda$'s parameters, yielding an estimator $\hat{\lambda}$, whose analysis is crucial for obtaining the main results. The main contributions include explicitly determining the convergence rates of the proposed estimator, which vary based on the estimation point. These results are compared to those for estimating the invariant density of a Levy process.
