On the nonparametric inference of coefficients of self-exciting jump-diffusion

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 long time horizon. We first propose to estimate the volatility coefficient. For that, we introduce in our estimation procedure a truncation function that allows to take into account the jumps of the process and we estimate the volatility function on a linear subspace of L 2 (A) where A is a compact interval of R. We obtain a bound for the empirical risk of the volatility estimator and establish an oracle inequality for the adaptive estimator to measure the performance of the procedure. Then, we propose an estimator of a sum between the volatility and the jump coefficient modified with the conditional expectation of the intensity of the jumps. The idea behind this is to recover the jump function. We also establish a bound for the empirical risk for the non-adaptive estimator of this sum and an oracle inequality for the final adaptive estimator. We conduct a simulation study to measure the accuracy of our estimators in practice and we discuss the possibility of recovering the jump function from our estimation procedure.