  In this paper, we consider a one-dimensional diffusion process with jumps
driven by a Hawkes process. We are interested in the estimations of the
volatility function and of the jump function from discrete high-frequency
observations in a long time horizon which remained an open question until now.
First, we propose to estimate the volatility coefficient. For that, we
introduce a truncation function in our estimation procedure that allows us to
take into account the jumps of the process and estimate the volatility function
on a linear subspace of L2(A) where A is a compact interval of R. We obtain a
bound for the empirical risk of the volatility estimator, ensuring its
consistency, and then we study an adaptive estimator w.r.t. the regularity.
Then, we define an estimator of a sum between the volatility and the jump
coefficient modified with the conditional expectation of the intensity of the
jumps. We also establish a bound for the empirical risk for the non-adaptive
estimators of this sum, the convergence rate up to the regularity of the true
function, and an oracle inequality for the final adaptive estimator.Finally, we
give a methodology to recover the jump function in some applications. We
conduct a simulation study to measure our estimators' accuracy in practice and
discuss the possibility of recovering the jump function from our estimation
procedure.
