Title: From Random Differential Equations to Structural Causal Models: the stochastic case,
Abstract: 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.