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

• URL: http://arxiv.org/abs/2607.12140v2
• Date: Wed, 22 Jul 2026 09:57:31 GMT
• Title: Causal Graphs, Markov Properties and Do-calculus for Stochastic Differential Equations
• Abstract summary: We consider systems of causal SDEs equipped with an explicit causal semantics.<n>For a class of additive-noise SDEs we prove a stronger $d$-separation Markov property, even if the system is cyclic.<n>We introduce time-split systems, which consider the causal relations between the processes when evaluated on disjoint intervals or time-points.
• Score: 0.6015898117103068
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $σ$-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.

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