Causal Graphs, Markov Properties and Do-calculus for Stochastic Differential Equations

Stochastic differential equations (SDEs) are widely used to model continuous-time dynamical systems, but graphical causal models for them are not yet well-understood.

We consider systems of causal SDEs that are equipped with an explicit causal semantics.

We pose solvability conditions for systems of causal SDEs such that they have well-defined observational and interventional distributions - even after marginalisation - and provide a general class of Lipschitz semimartingale SDEs that satisfies these conditions.

As core results we establish the $\sigma$-separation Markov property and the do-calculus in terms of the system's causal graph for probabilistic independence and interventions on the level of sample paths.

For a class of additive-noise SDEs we prove a stronger $d$-separation Markov property, even if the system is cyclic.

As a corollary of the do-calculus, we obtain an explicit causal interpretation of the graph: that the absence of a directed path implies the absence of a causal effect.

We further introduce time-split systems, which consider the causal relations between the processes when evaluated on disjoint intervals or time-points, and use them to reason about subsampled time-series, continuous-time Granger non-causality and local independence.

Finally, we discuss how constraint-based causal discovery algorithms (PC, FCI, CCD, CCI) apply directly to SDEs within our framework when conditional independence between sample paths can be consistently tested.