  Given discrete time observations over a growing time interval, we consider a
nonparametric Bayesian approach to estimation of the L\'evy density of a L\'evy
process belonging to a flexible class of infinite activity subordinators.
Posterior inference is performed via MCMC, and we circumvent the problem of the
intractable likelihood via the data augmentation device, that in our case
relies on bridge process sampling via Gamma process bridges. Our approach also
requires the use of a new infinite-dimensional form of a reversible jump MCMC
algorithm. We show that our method leads to good practical results in
challenging simulation examples. On the theoretical side, we establish that our
nonparametric Bayesian procedure is consistent: in the low frequency data
setting, with equispaced in time observations and intervals between successive
observations remaining fixed, the posterior asymptotically, as the sample size
$n\rightarrow\infty$, concentrates around the L\'evy density under which the
data have been generated. Finally, we test our method on a classical insurance
dataset.
