Abstract : We study the problem of optimal estimation of the jumps for stochastic processes. We assume that the stochastic process is discretely observed with a sampling step of size 1/n. We first propose an estimator of the sequence of jumps based on the discrete observations. This estimator has rate sqrt(n) and we give an explicit expression for its asymptotic error. Next, we show some lower bounds for the estimation of the jumps. When the marks of the underlying jump component are deterministic, we prove a LAMN property. We deduce then some convolution theorem, in the case where the marks of the jump component are random. Especially, this implies the optimality of our estimator.