论文标题
与布朗相关运动的扩散的非参数漂移估计
Nonparametric Drift Estimation from Diffusions with Correlated Brownian Motions
论文作者
论文摘要
在本文中，我们认为在$ [0，t] $上观察到$ n $扩散流程$ x^1，\ dots，x^n $，其中$ t $是固定的，而$ n $则成长为无限。与最近的大多数作品相反，我们不再认为这些过程是独立的。依赖性是通过驱动扩散过程的布朗运动之间的相关性来建模的。提出和研究了不使用相关矩阵知识的漂移函数的非参数估计器。它的集成平方风险是有限的，并提出了自适应程序。很少有理论上可以处理这种依赖性的工具，这使我们的结果成为新的。数值实验表明该过程在实践中起作用。
In the present paper, we consider that $N$ diffusion processes $X^1,\dots,X^N$ are observed on $[0,T]$, where $T$ is fixed and $N$ grows to infinity. Contrary to most of the recent works, we no longer assume that the processes are independent. The dependency is modeled through correlations between the Brownian motions driving the diffusion processes. A nonparametric estimator of the drift function, which does not use the knowledge of the correlation matrix, is proposed and studied. Its integrated mean squared risk is bounded and an adaptive procedure is proposed. Few theoretical tools to handle this kind of dependency are available, and this makes our results new. Numerical experiments show that the procedure works in practice.
