论文标题
随机阻尼哈密顿系统的固定分布的估计速率与连续观察
Rate of Estimation for the Stationary Distribution of Stochastic Damping Hamiltonian Systems with Continuous Observations
论文作者
论文摘要
我们研究了随机二维阻尼哈密顿系统$（z_t）_ {t \ in [0，t]} =（x_t，y_t，y_t，y_t）_ {t \ in [0，t]} $的固定分布的密度$π$的非参数估计问题。从$ [0，t] $上对采样路径的持续观察，我们研究了$π（x_0，y_0）$的估计速率为$ t \ to \ infty $。我们表明，基于内核的估计器可以实现（0,1/2）$的某些显式指数$ v \ $ t^{ - v} $。一个发现是估计速率取决于$π$的平滑度，并且与标准I.I.D. \ \设置中或二维非退化扩散过程中出现的速率完全不同。特别是，此速率也取决于$ y_0 $。此外，我们在$ l^2 $ - 风险上获得了最小的下限，以进行估计，其中相同的速率$ t^{ - v} $，最多$ \ log（t）$项。
We study the problem of the non-parametric estimation for the density $π$ of the stationary distribution of a stochastic two-dimensional damping Hamiltonian system $(Z_t)_{t\in[0,T]}=(X_t,Y_t)_{t \in [0,T]}$. From the continuous observation of the sampling path on $[0,T]$, we study the rate of estimation for $π(x_0,y_0)$ as $T \to \infty$. We show that kernel based estimators can achieve the rate $T^{-v}$ for some explicit exponent $v \in (0,1/2)$. One finding is that the rate of estimation depends on the smoothness of $π$ and is completely different with the rate appearing in the standard i.i.d.\ setting or in the case of two-dimensional non degenerate diffusion processes. Especially, this rate depends also on $y_0$. Moreover, we obtain a minimax lower bound on the $L^2$-risk for pointwise estimation, with the same rate $T^{-v}$, up to $\log(T)$ terms.
