  Uncertainty quantification plays an important role in problems that involve
inferring a parameter of an initial value problem from observations of the
solution. Conrad et al.\ (\textit{Stat.\ Comput.}, 2017) proposed randomisation
of deterministic time integration methods as a strategy for quantifying
uncertainty due to the unknown time discretisation error. We consider this
strategy for systems that are described by deterministic, possibly
time-dependent operator differential equations defined on a Banach space or a
Gelfand triple. Our main results are strong error bounds on the random
trajectories measured in Orlicz norms, proven under a weaker assumption on the
local truncation error of the underlying deterministic time integration method.
Our analysis establishes the theoretical validity of randomised time
integration for differential equations in infinite-dimensional settings.
