论文标题
离散观察到的扩散过程不变密度的估计：采样和异步性的影响
Estimation of the invariant density for discretely observed diffusion processes: impact of the sampling and of the asynchronicity
论文作者
论文摘要
我们旨在以非参数的方式估算A $ d $ d $二维的随机微分方程$（x_t）_ {t \ in [0，t]} $的密度$π$，for $ d \ ge 2 $，来自$ d \ ge 2 $，来自最有限的样本$ x_ $ x_ $ x_ $ x_ { t_0 <t_1 <... <t_n =：t_n $。我们提出了一个内核密度估计器，并研究了其在各向异性Hölder平滑度约束下对不变密度的次数估计的收敛速率。首先，我们在离散步骤中找到一些条件，以确保可以恢复相同的速率，就像该过程的连续轨迹一样。在密度估计器的背景下，此类速率是最佳的，并且是新的。然后，我们处理不满足离散步骤的这种条件的情况，我们将其称为中间制度。在这个新制度中，我们确定了各向异性Hölder类别估计不变密度的收敛速率，该类别的收敛速率与估计属于各向异性Hölder类的概率密度的估计相同，与$ n $ n $ iid Random Bandom Bastan Bastor类别相关。之后，我们专注于异步情况，其中可以在不同的时间点观察到每个组件。即使观测值的异步性使估算器方差的计算复杂化，我们也能够找到条件，以确保此方差与连续情况的一种相媲美。我们还展示了数据的非同步性在估计器研究中引入了其他偏差术语。
We aim at estimating in a non-parametric way the density $π$ of the stationary distribution of a $d$-dimensional stochastic differential equation $(X_t)_{t \in [0, T]}$, for $d \ge 2$, 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 estimator and we study its convergence rates for the pointwise estimation of the invariant density under anisotropic Hölder smoothness constraints. First of all, we find some conditions on the discretization step that ensures it is possible to recover the same rates as if the continuous trajectory of the process was available. 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 Hölder classes, which is the same convergence rate as for the estimation of a probability density belonging to an anisotropic Hölder 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 at different time points. Even if the asynchronicity of the observations complexifies the computation of the variance of the estimator, we are able to find conditions ensuring that this variance is comparable to the one of the continuous case. We also exhibit that the non synchronicity of the data introduces additional bias terms in the study of the estimator.
