  We perturb with an additive Gaussian white noise the Hamiltonian system
associated to a cubic anharmonic oscillator. The stochastic system is assumed
to start from initial conditions that guarantee the existence of a periodic
solution for the unperturbed equation. We write a formal expansion in powers of
the diffusion parameter for the candidate solution and analyze the
probabilistic properties of the sequence of the coefficients. It turns out that
such coefficients are the unique strong solutions of stochastic perturbations
of the famous Lam\'e's equation. We obtain explicit solutions in terms of
Jacobi elliptic functions and prove a lower bound for the probability that an
approximated version of the solution of the stochastic system stay close to the
solution of the deterministic problem. Conditions for the convergence of the
expansion are also provided.
