Optimal convergence rates for the invariant density estimation of jump-diffusion processes

We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for d = 1 and d = 2. We consider a class of jump diffusion processes whose invariant density belongs to some Hölder space. Firstly, in dimension one, we show that the kernel density estimator achieves the convergence rate 1/T, which is the optimal rate in the absence of jumps. This improves the convergence rate obtained in [Amorino, Gloter (2021)], which depends on the Blumenthal-Getoor index for d = 1 and is equal to log T/T for d = 2. Secondly, we show that is not possible to find an estimator with faster rates of estimation. Indeed, we get some lower bounds with the same rates { 1/T , log T/T } in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case.