  We consider a 1-dimensional diffusion process X with jumps. The particularity
of this model relies in the jumps which are driven by a multidimensional Hawkes
process denoted N. This article is dedicated to the study of a nonparametric
estimator of the drift coefficient of this original process. We construct
estimators based on discrete observations of the process X in a high frequency
framework with a large horizon time and on the observations of the process N.
The proposed nonparametric estimator is built from a least squares contrast
procedure on subspace spanned by trigonometric basis vectors. We obtain
adaptive results that are comparable with the one obtained in the nonparametric
regression context. We finally conduct a simulation study in which we first
focus on the implementation of the process and then on showing the good
behavior of the estimator.
