跳扩散过程不变密度估计的最优收敛速度
Optimal convergence rates for the invariant density estimation of jump-diffusion processes
摘要：
本文研究了一类低维带跳随机微分方程的不变密度估计问题，分别是$d=1$和$d=2$ .考虑一类跳扩散过程，其不变密度属于某个Hölder空间.首先，在一维，我们证明了核密度估计达到收敛速度$\frac{1}{T}$，这是在没有跳跃的情况下的最佳速度。这提高了[Amorino，Gloter（2021）]中获得的收敛速度，其取决于$d=1$的Blumenthal-Getoor指数，并且等于$d=2$的$\frac{\log T}{T}$。其次，我们表明，是不可能找到一个估计速度更快的估计。事实上，我们得到了一些下界相同的速率$\{\frac{1}{T},\frac{\log T}{T}\}$在单，二维的情况下，分别。最后，我们得到了一维情形下估计量的渐近正态性。
英文摘要：
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 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 $\frac{1}{T}$, which is the optimal rate in the absence of jumps. This improves the convergence rate obtained in [Amorino, Gloter (2021)], which depends on the Blumenthal-Getoor index for $d=1$ and is equal to $\frac{\log T}{T}$ for $d=2$. Secondly, 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 $\{\frac{1}{T},\frac{\log T}{T}\}$ in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case.
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