Stochastic volatility and jumps are viewed as arising from Brownian subordination
given here by an independent purely discontinuous process and we inquire into the
relation between the realized variance or quadratic variation of the process and
the time change. The class of models considered encompasses a wide range of models
employed in practical financial modeling. It is shown that in general the time change
cannot be recovered from the composite process and we obtain its conditional distribution
in a variety of cases. The implications of our results for working with stochastic
volatility models in general is also described. We solve the recovery problem, i.e.
the identification the conditional law for a variety of cases, the simplest solution
being for the gamma time change when this conditional law is that of the first hitting
time process of Brownian motion with drift attaining the level of the variation of
the time changed process. We also introduce and solve in certain cases the problem
of stochastic scaling. A stochastic scalar is a subordinator that recovers the law
of a given subordinator when evaluated at an independent and time scaled copy of
the given subordinator. These results are of importance in comparing price quality
delivered by alternate exchanges.