  The parameterization method (PM) provides a broad theoretical and numerical
foundation for computing invariant manifolds of dynamical systems. PM
implements a change of variables in order to represent trajectories of a system
of ordinary differential equations ``as simply as possible." In this paper we
pursue a similar goal for stochastic oscillator systems. For planar nonlinear
stochastic systems that are ``robustly oscillatory", we find a change of
variables through which the dynamics are as simple as possible $\textit{in the
mean}$. We prove existence and uniqueness of a deterministic vector field, the
trajectories of which capture the local mean behavior of the stochastic
oscillator. We illustrate the construction of such an ``effective vector field"
for several examples, including a limit cycle oscillator perturbed by noise, an
excitable system derived from a spiking neuron model, and a spiral sink with
noise forcing (2D Ornstein-Uhlenbeck process). The latter examples comprise
contingent oscillators that would not sustain rhythmic activity without noise
forcing. Finally, we exploit the simplicity of the dynamics after the change of
variables to obtain the effective diffusion constant of the resulting phase
variable, and the stationary variance of the resulting amplitude (isostable)
variable.
