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 fully non-linear 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 and Gloter [J. Stat. Plann. Inference 213 (2021) 106-129], which depends on the Blumenthal-Getoor index for d = 1 and is equal to (logT)/T for d = 2. Secondly, when the jump and diffusion coefficients are constant and the jumps are finite, 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, (logT)/T} in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case for the fully non-linear process.