  We address the calculation of transition probabilities in multiplicative
noise stochastic differential equations using a path integral approach. We show
the equivalence between the conditional probability and the propagator of a
quantum particle with variable mass. Introducing a {\em time
reparametrization}, we are able to transform the problem of multiplicative
noise fluctuations into an equivalent additive one. We illustrate the method by
showing the explicit analytic computation of the conditional probability of a
harmonic oscillator in a nonlinear multiplicative environment.
