论文标题
Malliavin conculus，以最佳估计不变的密度在中间状态中的扩散膨胀
Malliavin calculus for the optimal estimation of the invariant density of discretely observed diffusions in intermediate regime
论文作者
论文摘要
令$（x_t）_ {t \ ge 0} $为一维随机微分方程的解决方案。我们的目的是研究中级政权中不变密度的收敛速率，假设对流程的离散观察$（x_t）_ {t \ in [0，t]} $，当$ t $倾向于$ \ infty $。我们发现与我们提出的内核密度估计器相关的收敛速率以及对离散步骤$δ_n$的条件，该条件在中间式和连续情况下起着阈值的作用。在中级方案中，收敛速率为$ n^{ - \ frac {2β} {2β+ 1}}} $，其中$β$是不变密度的平滑度。之后，我们补充了先前发现的上限与所有可能的估计器集合的下限，这提供了相同的收敛速率：这意味着不可能提出一个不同的估计器，该估计量可以实现更好的收敛速率。这是通过两个假设方法获得的。最具挑战性的部分是在两个模型的定律之间界定地狱灵魂的距离。关键点是分数函数的Malliavin表示，这使我们能够根据Malliavin的重量来绑定Hellinger距离。
Let $(X_t)_{t \ge 0}$ be solution of a one-dimensional stochastic differential equation. Our aim is to study the convergence rate for the estimation of the invariant density in intermediate regime, assuming that a discrete observation of the process $(X_t)_{t \in [0, T]}$ is available, when $T$ tends to $\infty$. We find the convergence rates associated to the kernel density estimator we proposed and a condition on the discretization step $Δ_n$ which plays the role of threshold between the intermediate regime and the continuous case. In intermediate regime the convergence rate is $n^{- \frac{2 β}{2 β+ 1}}$, where $β$ is the smoothness of the invariant density. After that, we complement the upper bounds previously found with a lower bound over the set of all the possible estimator, which provides the same convergence rate: it means it is not possible to propose a different estimator which achieves better convergence rates. This is obtained by the two hypotheses method; the most challenging part consists in bounding the Hellinger distance between the laws of the two models. The key point is a Malliavin representation for a score function, which allows us to bound the Hellinger distance through a quantity depending on the Malliavin weight.
