  In this work we consider the unbiased estimation of expectations
w.r.t.~probability measures that have non-negative Lebesgue density, and which
are known point-wise up-to a normalizing constant. We focus upon developing an
unbiased method via the underdamped Langevin dynamics, which has proven to be
popular of late due to applications in statistics and machine learning.
Specifically in continuous-time, the dynamics can be constructed {so that as
the time goes to infinity they} admit the probability of interest as a
stationary measure. {In many cases, time-discretized versions of the
underdamped Langevin dynamics are used in practice which are run only with a
fixed number of iterations.} We develop a novel scheme based upon doubly
randomized estimation as in \cite{ub_grad,disc_model}, which requires access
only to time-discretized versions of the dynamics. {The proposed scheme aims to
remove the dicretization bias and the bias resulting from running the dynamics
for a finite number of iterations}. We prove, under standard assumptions, that
our estimator is of finite variance and either has finite expected cost, or has
finite cost with a high probability. To illustrate our theoretical findings we
provide numerical experiments which verify our theory, which include
challenging examples from Bayesian statistics and statistical physics.
