In this paper, we present conditions for identifying the generator of a linear stochastic differential equation (SDE) from the distribution of its solution process with a given fixed initial state. These identifiability conditions are crucial in causal inference using linear SDEs as they enable the identification of the post-intervention distributions from its observational distribution. Specifically, we derive a sufficient and necessary condition for identifying the generator of linear SDEs with additive noise, as well as a sufficient condition for identifying the generator of linear SDEs with multiplicative noise. We show that the conditions derived for both types of SDEs are generic. Moreover, we offer geometric interpretations of the derived identifiability conditions to enhance their understanding. To validate our theoretical results, we perform a series of simulations, which support and substantiate the established findings.

翻译：本文提出了从给定固定初始状态的解过程分布中识别线性随机微分方程生成元的条件。这些可识别性条件对于利用线性SDE进行因果推断至关重要，因为它们使得从观测分布中识别干预后分布成为可能。具体而言，我们推导了带加性噪声的线性SDE生成元可识别的充要条件，以及带乘性噪声的线性SDE生成元可识别的充分条件。我们证明两类SDE所推导的条件具有普适性。此外，我们为推导出的可识别性条件提供几何解释以加深理解。为验证理论结果，我们进行了一系列仿真实验，这些实验支持并证实了所建立的结论。