  We consider continuous-time survival or more general event-history settings,
where the aim is to infer the causal effect of a time-dependent treatment
process. This is formalised as the effect on the outcome event of a (possibly
hypothetical) intervention on the intensity of the treatment process, i.e. a
stochastic intervention. To establish whether valid inference about the
interventional situation can be drawn from typical observational, i.e.
non-experimental, data we propose graphical rules indicating whether the
observed information is sufficient to identify the desired causal effect by
suitable re-weighting. In analogy to the well-known causal directed acyclic
graphs, the corresponding dynamic graphs combine causal semantics with local
independence models for multivariate counting processes. Importantly, we
highlight that causal inference from censored data requires structural
assumptions on the censoring process beyond the usual independent censoring
assumption, which can be represented and verified graphically. Our results
establish general non-parametric identifiability and do not rely on particular
survival models. We illustrate our proposal with a data example on HPV-testing
for cervical cancer screening, where the desired effect is estimated by
re-weighted cumulative incidence curves.
