We extend Robins’ theory of causal inference for complex longitudinal data to the case of continuously varying as opposed to discrete covariates and treatments. In particular we establish versions of the key results of the discrete theory: the $g$-computation formula and a collection of powerful characterizations of the $g$-null hypothesis of no treatment effect. This is accomplished under natural continuity hypotheses concerning the conditional distributions of the outcome variable and of the covariates given the past. We also show that our assumptions concerning counterfactual variables place no restriction on the joint distribution of the observed variables: thus in a precise sense, these assumptions are “for free,” or if you prefer, harmless.
"Causal Inference for Complex Longitudinal Data: The Continuous Case." Ann. Statist. 29 (6) 1785 - 1811, December 2001. https://doi.org/10.1214/aos/1015345962