First, we consider a stochastic differential equation with jumps driven by a finite activity Lévy process and we assume that the jump amplitudes have a centered density with finite moments. We show upper and lower estimates for the density of the solution in the case that the jump amplitudes follow a Gaussian or Laplacian law. The proof of the lower bound uses a general expression for the density of the solution. The proof of the upper bound uses techniques of the Malliavin calculus.

Second, we consider a Vasicek model with jumps driven by a Lévy process and a Cox-Ingersoll-Ross process with jumps driven by a subordinator, whose drift coefficients depend on unknown parameters. We show the Local Asymptotic Normality (LAN), Local Asymptotic Mixed Normality (LAMN) and Local Asymptotic Quadraticity (LAQ) properties for these models in both ergodic and non-ergodic cases when the processes are observed discretely at high frequency over an increasing time interval. Furthermore, we prove the LAMN property for the diffusion parameter of stochastic differential equations with jumps driven by a finite activity Lévy process when the solution process is observed discretely at high frequency over a fixed time interval. The proofs are essentially based on Malliavin calculus techniques and a subtle analysis on the jump structure of the Lévy process.

Third, we propose a tamed-adaptive Euler-Maruyama approximation scheme for Lévy-driven stochastic differential equations with locally Lipschitz continuous, polynomial growth drift, and locally Holder continuous, polynomial growth diffusion coefficients. We prove that the new scheme converges in both finite and infinite time intervals under some suitable conditions on the regularity and the growth of the coefficients and the integrability of the Lévy measure.

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