Non parametric estimation of the jump coefficient of a diffusion with jumps

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