Rate of estimation for the stationary distribution of jump-processes over anisotropic Holder classes

Abstract

We study the problem of the non-parametric estimation for the density π of the stationary distribution of the multivariate stochastic differential equation with jumps (Xt) 0≤t≤T , when the dimension d is such that d ≥ 3. From the continuous observation of the sampling path on [0, T ] we show that, under anisotropic Hölder smoothness constraints, kernel based estimators can achieve fast convergence rates. In particular, they are as fast as the ones found by Dalalyan and Reiss [11] for the estimation of the invariant density in the case without jumps under isotropic Hölder smoothness constraints. Moreover, they are faster than the ones found by Strauch [32] for the invariant density estimation of continuous stochastic differential equations, under anisotropic Hölder smoothness constraints. Furthermore, we obtain a minimax lower bound on the L2-risk for pointwise estimation, with the same rate up to a log(T ) term. It implies that, on a class of diffusions whose invariant density belongs to the anisotropic Holder class we are considering, it is impossible to find an estimator with a rate of estimation faster than the one we propose.

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

“…The proof of Theorem 3 follows along the same lines as that of Theorem 2 in [3], where a lower bound for the kernel estimator of the invariant density for the solution to (8) for d ≥ 3 is obtained. The proof is based on the two hypotheses method, explained for example in Section 2.3 of [37].…”

“…Instead, we use the Kullback's version of the finite number of hypotheses method as stated in Lemma C.1 of [36], see Lemma 2 below. Observe that this method gives a slightly weaker lower bound as we get a sup x inside the expectation, while the method in [3] provides an inf x outside the expectation.…”

“…After that, we will discuss the optimality of the convergence rate in the bi-dimensional case. This will close the circle of the analysis of the convergence rates for the estimation of the invariant density of jump-diffusions, as the convergence rates and their optimality in the case d ≥ 3 have already been treated in detail in [3].…”

“…The same framework has been considered in some recent papers such as [2], [23] (Section 5.2), and [3]. In the first paper, it is shown that the kernel estimator achieves the following convergence rates for the pointwise estimation of the invariant density: log T T for d = 2 and (log T ) (2− (1+α) 2…”

“…If in absence of jumps such link is explicit, in our context it is more challenging. As shown in [19] and [3], it is possible to find the link knowing that the invariant measure has to satisfy A * µ b = 0, where A * is the adjoint of the generator of the considered diffusion. This method allows us to show that the superoptimal rate 1 T is the best possible for the estimation of the invariant density in d = 1, but it fails in the bi-dimensional case (see Remark 1 below for details).…”