  Classical nucleation theory (CNT) is the most widely used framework to
describe the early stage of first-order phase transitions. Unfortunately the
different points of view adopted to derive it yield different kinetic equations
for the probability density function, e.g. Zeldovich-Frenkel or
Becker-D\"oring-Tunitskii equations. Starting from a phenomenological
stochastic differential equation a unified equation is obtained in this work.
In other words, CNT expressions are recovered by selecting one or another
stochastic calculus. Moreover, it is shown that the unified CNT thus obtained
produces the same Fokker-Planck equation as that from a recent update of CNT
[J.F. Lutsko and M.A. Dur\'an-Olivencia, J. Chem. Phys., 2013, 138, 244908]
when mass transport is governed by diffusion. Finally, we derive a general
induction-time expression along with specific approximations of it to be used
under different scenarios. In particular, when the mass-transport mechanism is
governed by direct impingement, volume diffusion, surface diffusion or
interface transfer.
