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：本文给出从具有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（SDE）发生元的条件。这些可辨识条件在使用线性SDE的因果推断中至关重要，因为它们能够从观测分布中识别干预后分布。具体而言，我们推导了加性噪声线性SDE发生元可辨识的充要条件，以及乘性噪声线性SDE发生元可辨识的充分条件。我们证明这两种类型SDE推导出的条件具有普适性。此外，我们给出所推导可辨识条件的几何解释以增强理解。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。