  Quantile clocks are defined as convolutions of subordinators $L$, with
quantile functions of positive random variables. We show that quantile clocks
can be chosen to be strictly increasing and continuous and discuss their
practical modeling advantages as business activity times in models for asset
prices. We show that the marginal distributions of a quantile clock, at each
fixed time, equate with the marginal distribution of a single subordinator.
Moreover, we show that there are many quantile clocks where one can specify
$L$, such that their marginal distributions have a desired law in the class of
generalized $s$-self decomposable distributions, and in particular the class of
self-decomposable distributions. The development of these results involves
elements of distribution theory for specific classes of infinitely divisible
random variables and also decompositions of a gamma subordinator, that is of
independent interest. As applications, we construct many price models that have
continuous trajectories, exhibit volatility clustering and have marginal
distributions that are equivalent to those of quite general exponential
L\'{e}vy price models. In particular, we provide explicit details for
continuous processes whose marginals equate with the popular VG, CGMY and NIG
price models. We also show how to perfectly sample the marginal distributions
of more general classes of convoluted subordinators when $L$ is in a sub-class
of generalized gamma convolutions, which is relevant for pricing of European
style options.
