  Doubly robust estimators have gained widespread popularity in various fields
due to their ability to provide unbiased estimates under model
misspecification. However, the asymptotic theory for doubly robust estimators
with continuous-time nuisance parameters remains largely unexplored. In this
short communication, we address this gap by developing a general asymptotic
theory for a class of doubly robust estimating equations involving stochastic
processes and Riemann-Stieltjes integrals. We introduce generic assumptions on
the nuisance parameter estimators that ensure the consistency and asymptotic
normality of the resulting doubly robust estimator. Our results cover both the
model doubly robust estimator, which relies on parametric or semiparametric
models, and the rate doubly robust estimator, which allows for flexible machine
learning methods. We discuss the implications of our findings and highlight the
key differences between the continuous-time setting and the classical theory
for doubly robust estimators. Our work provides a solid theoretical foundation
for the use of doubly robust estimators in complex settings with
continuous-time nuisance parameters, paving the way for future research and
applications.
