  Aiming at enlarging the class of symmetries of an SDE, we introduce a family
of stochastic transformations able to change also the underlying probability
measure exploiting Girsanov Theorem and we provide new determining equations
for the infinitesimal symmetries of the SDE. The well-defined subset of the
previous class of measure transformations given by Doob transformations allows
us to recover all the Lie point symmetries of the Kolmogorov equation
associated with the SDE. This gives the first stochastic interpretation of all
the deterministic symmetries of the Kolmogorov equation. The general theory is
applied to some relevant stochastic models.
