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

2026-07-13

AI Summary

Existing graphical causal models struggle to represent the causal structure of continuous-time dynamical systems governed by stochastic differential equations (SDEs), particularly lacking theoretical foundations regarding interventional distributions, marginalization, and sample-path properties. This work establishes a rigorous semantic framework for causal SDEs, introducing solvability conditions that ensure well-defined observational and interventional distributions under marginalization. It formally develops σ-separation-based Markov properties and do-calculus rules within the SDE setting for the first time. For additive-noise SDEs, the paper proves that a stronger d-separation property holds even in cyclic systems. A time-rescaling transformation is introduced to unify subsampling, Granger non-causality, and local independence. These results demonstrate that constraint-based causal discovery algorithms—such as PC, FCI, CCD, and CCI—can be directly applied in this continuous-time framework.

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.

Problem

• Stochastic Differential Equations
• Causal Graphs
• Markov Properties
• Do-calculus
• Causal Inference

Innovation

• causal SDEs
• σ-separation
• do-calculus
• time-split systems
• Markov property