Nonparametric Estimation for Hawkes-Diffusion Systems

par Charlotte Dion-Blanc

mardi 9 déc. 2025, 14:00 → 15:00 Europe/Paris

In this talk, I will study two models based on point processes and stochastic differential equations (SDEs). Both models are motivated by neuroscience applications, which I will briefly present.

In the first part, I will study a diffusion process with jumps driven by a Hawkes process, where the intensity of the Hawkes process is a piecewise deterministic Markov process. After presenting the asymptotic properties of the system, I will focus on estimating the invariant density of the joint process formed by the diffusion component and the stochastic intensity process. This step is essential due to the potential applications of the model. Our rates of convergence are compared in particular with those obtained for the estimation of the invariant density of a Lévy process.

In the second part of the talk, I will study a Hawkes process with a baseline driven by an SDE. I will again present the asymptotic behavior of the process, but then focus on the maximum likelihood estimator. The intensity process is assumed to depend on an unknown parameter, which we prove to be asymptotically normal as the observation time goes to infinity.