We consider the problem of estimating unknown parameters in stochastic differential equations driven by colored noise, given continuous-time observations. Colored noise is modelled as a sequence of mean zero Gaussian stationary processes with an exponential autocorrelation function, with decreasing correlation time. Our goal is to infer parameters in the limit equation, driven by white noise, given observations of the colored noise dynamics. As in the case of parameter estimation for multiscale diffusions, the observations are only compatible with the data in the white noise limit, and classic estimators become biased, implying the need of preprocessing the data. We consider both the maximum likelihood and the stochastic gradient descent in continuous time estimators, and we propose modified versions of these methods, in which the observations are filtered using an exponential filter. Both stochastic differential equations with additive and multiplicative noise are considered. We provide a convergence analysis for our novel estimators in the limit of infinite data, and in the white noise limit, showing that the estimators are asymptotically unbiased. We consider in detail the case of multiplicative colored noise, in particular when the L\'evy area correction drift appears in the limiting white noise equation. A series of numerical experiments corroborates our theoretical results.

翻译：我们考虑在连续时间观测下，由色噪声驱动的随机微分方程中未知参数的估计问题。色噪声建模为具有指数自相关函数（相关时间递减）的零均值高斯平稳过程序列。我们的目标是在已知色噪声动力学观测数据的情况下，推断由白噪声驱动的极限方程中的参数。与多尺度扩散参数估计问题类似，观测数据仅在与白噪声极限兼容时才有效，而经典估计量会产生偏差，因此需要对数据进行预处理。我们同时考虑了连续时间下的极大似然估计和随机梯度下降法，并提出了这两种方法的修正版本——使用指数滤波器对观测数据进行滤波。本文分别研究了加性噪声和乘性噪声驱动的随机微分方程。针对新提出的估计量，我们在无限数据极限和白噪声极限下进行了收敛性分析，证明估计量是渐近无偏的。我们重点分析了乘性色噪声情形，特别是当Lévy面积修正漂移项出现在极限白噪声方程中的情况。一系列数值实验验证了我们的理论结果。