We consider the problem of estimating unknown parameters in stochastic differential equations driven by colored noise, which we model as a sequence of Gaussian stationary processes with decreasing correlation time. We aim to infer parameters in the limit equation, driven by white noise, given observations of the colored noise dynamics. We consider both the maximum likelihood and the stochastic gradient descent in continuous time estimators, and we propose to modify them by including filtered data. We provide a convergence analysis for our estimators showing their asymptotic unbiasedness in a general setting and asymptotic normality under a simplified scenario.

翻译：我们考虑由色噪声驱动的随机微分方程中未知参数的估计问题，其中色噪声被建模为一系列相关时间递减的高斯平稳过程。我们的目标是根据色噪声动态观测数据，推断由白噪声驱动的极限方程中的参数。我们考虑了连续时间估计量中的极大似然估计和随机梯度下降法，并提出通过引入滤波数据对其进行改进。我们提供了所提出估计量的收敛性分析，证明其在一般设定下的渐近无偏性，并在简化场景下证明了其渐近正态性。