  A description in terms of phase and amplitude variables is given, for
nonlinear oscillators subject to white Gaussian noise described by It\^o
stochastic differential equations. The stochastic differential equations
derived for the amplitude and the phase are rigorous, and their validity is not
limited to the weak noise limit. If the noise intensity is small, the equations
can be efficiently solved using asymptotic expansions. Formulas for the
expected angular frequency, expected oscillation amplitude and amplitude
variance are derived using It\^o calculus.
