Abstract
We give a causal interpretation of stochastic differential equations (SDEs) by defining the postintervention SDE resulting from an intervention in an SDE. We show that under Lipschitz conditions, the solution to the postintervention SDE is equal to a uniform limit in probability of postintervention structural equation models based on the Euler scheme of the original SDE, thus relating our definition to mainstream causal concepts. We prove that when the driving noise in the SDE is a L\'evy process, the postintervention distribution is identifiable from the generator of the SDE.
Highlights
The notion of causality has long been of interest to both statisticians and scientists working in fields applying statistics
(2) We show that under certain regularity assumptions, the solution of the postintervention stochastic differential equations (SDEs) is the limit of a sequence of interventions in structural equation models based on the Euler scheme of the observational SDE
(3) We prove using (2) that for SDEs with a Levy process as the driving semimartingale, the postintervention distribution is identifiable from the generator associated with the SDE
Summary
The notion of causality has long been of interest to both statisticians and scientists working in fields applying statistics. Theorem 5.3 essentially shows that for Levy driven SDE models, the effect of interventions can be uniquely identified from the observational distribution, meaning that the intervention effect identification problem present in classical DAG-based models vanishes for these SDE models We expect that this result will have considerable applicability for causal inference for time-dependent observations. Given an SDE model, in order to use the definition of postintervention SDEs given here to predict the effects of real-world interventions, it is necessary that the SDE can be sensibly interpreted as a data-generating mechanism with certain properties: as we will argue, it is essentially sufficient that the driving semimartingales are autonomous in the sense that they may be assumed not to be directly affected by interventions This is an assumption which is not testable from a statistical viewpoint.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
