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In this paper, we present conditions for identifying the generator of alinear stochastic differential equation (SDE) from the distribution of itssolution process with a given fixed initial state. These identifiabilityconditions are crucial in causal inference using linear SDEs as they enable theidentification of the post-intervention distributions from its observationaldistribution. Specifically, we derive a sufficient and necessary condition foridentifying the generator of linear SDEs with additive noise, as well as asufficient condition for identifying the generator of linear SDEs withmultiplicative noise. We show that the conditions derived for both types ofSDEs are generic. Moreover, we offer geometric interpretations of the derivedidentifiability conditions to enhance their understanding. To validate ourtheoretical results, we perform a series of simulations, which support andsubstantiate the established findings.

Generator Identification for Linear SDEs with Additive and Multiplicative Noise