LAN property for diffusion processes with jumps with discrete observations via Malliavin's calculation.

Summary In this thesis we apply Malliavin's calculus to obtain the local asymptotic normality (LAN) property from discrete observations of certain uniformly elliptic diffusion processes with jumps. In Chapter 2 we revise the proof of the local asymptotic mixed normality (LAMN) property for diffusion processes with jumps from continuous observations, and as a consequence we obtain the LAN property assuming the ergodicity of the process. In Chapter 3 we establish the LAN property for a simple Lévy process with unknown drift and diffusion parameters and intensity. In Chapter 4, using Malliavin's calculation and transition density estimates, we prove that the LAN property is verified for a jumping diffusion process whose drift coefficient depends on an unknown parameter. Finally, in the same direction we obtain in Chapter 5 the LAN property for a jump diffusion process where the two unknown parameters are involved in the drift and diffusion coefficients.