Non parametric estimation of the diffusion coefficents of a diffusion with jumps

In this article, we consider a jump diffusion process (X_t), with drift function b, diffusion coefficient sigma and jump coefficient xi^{2}. This process is observed at discrete times t=0,Delta,...,nDelta. The sampling interval Delta tends to 0 and nDelta tends to infinity. We assume that (X_t) is ergodic, strictly stationary and exponentially beta-mixing. We use a penalized least-square approach to compute adaptive estimators of the functions sigma^2+xi^2 and sigma^2. We provide bounds for the risks of the two estimators.