Estimation of the invariant density for discretely observed diffusion processes: impact of the sampling and of the asynchronicity
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We aim at estimating in a non-parametric way the density π of the stationary distribution of a d-dimensional stochastic differential equation (X_t)_t ∈ [0, T], for d ≥ 2, from the discrete observations of a finite sample X_t_0, ... , X_t_n with 0= t_0 < t_1 < ... < t_n =: T_n. We propose a kernel density estimation and we study its convergence rates for the pointwise estimation of the invariant density under anisotropic Holder smoothness constraints. First of all, we find some conditions on the discretization step that ensures it is possible to recover the same rates as when the continuous trajectory of the process was available. As proven in the recent work <cit.>, such rates are optimal and new in the context of density estimator. Then we deal with the case where such a condition on the discretization step is not satisfied, which we refer to as intermediate regime. In this new regime we identify the convergence rate for the estimation of the invariant density over anisotropic Holder classes, which is the same convergence rate as for the estimation of a probability density belonging to an anisotropic Holder class, associated to n iid random variables X_1, ..., X_n. After that we focus on the asynchronous case, in which each component can be observed in different moments. Even if the asynchrony of the observations implies some difficulties, we are able to overcome them by considering some combinatorics and by proving some sharper bounds on the variance which allow us to lighten the condition needed on the discretization step.
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