论文标题
在各向异性类别上的不变性跳跃扩散过程的不变密度自适应估计
Invariant density adaptive estimation for ergodic jump diffusion processes over anisotropic classes
论文作者
论文摘要
我们考虑X =（XT）t $ \ ge $ 0的多元随机微分方程，具有征收型跳跃，并具有密度$μ$的独特不变概率度量。我们假设有连续的观察记录x t =（xt）0 $ \ le $ t $ \ le $ t。在没有跳跃的情况下，Reiss和Dalalyan（2007）和Strauch（2018）发现，在各向同性和各向异性H {Ö} lder平滑度约束下，分别不变密度估计器的收敛速率，它们比从标准多变量密度估计中知道的速度要快得多。我们通过在有跳跃的情况下获得一些估计量，而这些估计量与d $ \ ge $ 2的相同的融合率和相同的收敛速率相同，而汇率则取决于一维设置中的跳跃程度。此外，我们提出了一个基于Goldensh-Luger和Lepski（2011）方法的数据驱动带宽选择程序，该方法导致我们进行了自适应的非参数核心估计器，对固定密度$μ$的跳跃扩散x的均值x.适应性带宽选择，anisotropic strivion vily vys vys vys vy sive vy sive，
We consider the solution X = (Xt) t$\ge$0 of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density $μ$. We assume that a continuous record of observations X T = (Xt) 0$\le$t$\le$T is available. In the case without jumps, Reiss and Dalalyan (2007) and Strauch (2018) have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic H{ö}lder smoothness constraints, which are considerably faster than those known from standard multivariate density estimation. We extend the previous works by obtaining, in presence of jumps, some estimators which have the same convergence rates they had in the case without jumps for d $\ge$ 2 and a rate which depends on the degree of the jumps in the one-dimensional setting. We propose moreover a data driven bandwidth selection procedure based on the Goldensh-luger and Lepski (2011) method which leads us to an adaptive non-parametric kernel estimator of the stationary density $μ$ of the jump diffusion X. Adaptive bandwidth selection, anisotropic density estimation, ergodic diffusion with jumps, L{é}vy driven SDE
