Causal interpretation of stochastic differential equations

TL;DR

This paper introduces a causal framework for stochastic differential equations (SDEs), defining postintervention SDEs and establishing their relation to structural models, with results on identifiability when driven by Lévy processes.

Contribution

It provides a novel causal interpretation of SDEs, linking interventions to solutions of postintervention SDEs and connecting to existing structural equation models.

Findings

Postintervention SDE solutions converge to structural equation models.

Postintervention distribution is identifiable from the generator when driven by Lévy processes.

The approach relates causal concepts to stochastic differential equations.

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.