From Random Differential Equations to Structural Causal Models: the stochastic case

Random Differential Equations provide a natural extension of Ordinary Differential Equations to the stochastic setting. We show how, and under which conditions, every equilibrium state of a Random Differential Equation (RDE) can be described by a Structural Causal Model (SCM), while pertaining the causal semantics. This provides an SCM that captures the stochastic and causal behavior of the RDE, which can model both cycles and confounders. This enables the study of the equilibrium states of the RDE by applying the theory and statistical tools available for SCMs, for example, marginalizations and Markov properties, as we illustrate by means of an example. Our work thus provides a direct connection between two fields that so far have been developing in isolation.