基于连续观测的分数阶扩散模型中的随机效应估计
Random effects estimation in a fractional diffusion model based on continuous observations
AI总结：
本研究针对分数阶扩散模型的随机效应，结合最大似然参数估计与Bernstein多项式非参数逼近构建混合估计方法，推导其渐近性质，并通过数值模拟验证其密度估计性能优于标准核估计。
摘要：
本研究的目的是采用混合估计方法构建分数阶扩散模型中随机效应的估计量，该方法结合了参数与非参数技术。我们精确考虑了在时间区间[0,T]上被连续观测的n个随机过程{X_t^j, 0≤t≤T}，j=1,…,n，其中每个过程的动力学由漂移项依赖于随机效应的分数阶随机微分方程描述。我们首先利用最大似然估计技术构建了随机效应的参数估计量，并研究了当时间跨度T足够大时该估计量的渐近性质。随后，结合已得到的随机效应估计量，我们使用Bernstein多项式逼近构建了其共同未知密度函数的非参数估计量。针对无限时间跨度T和固定样本量n的情况，我们研究了该密度估计量的若干渐近性质，例如其渐近偏差、方差和平均积分平方误差。针对无限时间跨度T、高频情形以及Bernstein多项式阶数足够大的情况，我们探究了该估计量的渐近正态性和一致收敛性。我们还开展了若干数值模拟，以说明相较于标准Kernel估计量，基于Bernstein多项式的估计量在估计随机效应密度函数时的性能表现。
英文摘要：
The purpose of the present work is to construct estimators for the random effects in a fractional diffusion model using a hybrid estimation method where we combine parametric and nonparametric thechniques. We precisely consider $n$ stochastic processes $\left\{X_t^j,\ 0\leq t\leq T\right\}$, $j=1,\ldots, n$ continuously observed over the time interval $[0,T]$, where the dynamics of each process are described by fractional stochastic differential equations with drifts depending on random effects. We first construct a parametric estimator for the random effects using the techniques of maximum likelihood estimation and we study its asymptotic properties when the time horizon $T$ is sufficiently large. Then by taking into account the obtained estimator for the random effects, we build a nonparametric estimator for their common unknown density function using Bernstein polynomials approximation. Some asymptotic properties of the density estimator, such as its asymptotic bias, variance and mean integrated squared error, are studied for an infinite time horizon $T$ and a fixed sample size $n$. The asymptotic normality and the uniform convergence of the estimator are investigated for an infinite time horizon $T$, a high frequency and as the order of Bernstein polynomials is sufficiently large. Some numerical simulations are also presented to illustrate the performance of the Bernstein polynomials based estimator compared to standard Kernel estimator for the random effects density function.
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