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.

翻译：本文考虑在固定时间区间$[0,T]$上观测到$N$个扩散过程$X^1,\dots,X^N$的情形，其中$T$固定而$N$趋于无穷。与近期大多数研究不同，我们不再假设这些过程相互独立。过程间的依赖性通过驱动扩散过程的布朗运动之间的相关性来建模。本文提出并研究了一种无需知晓相关矩阵的漂移函数非参数估计量，证明了其积分均方风险有界，并提出了自适应估计方法。由于处理此类依赖关系的理论工具十分有限，我们的结果具有创新性。数值实验表明该方法在实际中有效。