  Many real-world processes are trajectories that may be regarded as
continuous-time "functional data". Examples include patients' biomarker
concentrations, environmental pollutant levels, and prices of stocks.
Corresponding advances in data collection have yielded near continuous-time
measurements, from e.g. physiological monitors, wearable digital devices, and
environmental sensors. Statistical methodology for estimating the causal effect
of a time-varying treatment, measured discretely in time, is well developed.
But discrete-time methods like the g-formula, structural nested models, and
marginal structural models do not generalize easily to continuous time, due to
the entanglement of uncountably infinite variables. Moreover, researchers have
shown that the choice of discretization time scale can seriously affect the
quality of causal inferences about the effects of an intervention. In this
paper, we establish causal identification results for continuous-time
treatment-outcome relationships for general cadlag stochastic processes under
continuous-time confounding, through orthogonalization and weighting. We use
three concrete running examples to demonstrate the plausibility of our
identification assumptions, as well as their connections to the discrete-time g
methods literature.
