In this article, we consider a jump diffusion process \((X_t)_{t \ge 0}\) with drift function b, diffusion coefficient \(\sigma \) and jump coefficient \(\xi \) . This process is supposed to be ergodic, exponentially \(\beta \) -mixing and stationary. It is observed at discrete times \(t=0,\Delta ,\ldots ,n\Delta \) . The sampling interval \(\Delta \) tends to 0 and the time interval \(n\Delta \) tends to infinity. We construct a robust, adaptive non-parametric estimator of the function \(\xi ^4\) thanks to a penalized least-square approach. We provide bounds of the empirical and \(L^2\) -risk of our estimator.
