  Parametric rolling is one of the dangerous dynamic phenomena. In order to
discuss the safety of a vessel when a dangerous phenomenon occurs, it is
important to estimate the probability of certain dynamical behavior of the ship
with respect to a certain threshold level. In this paper, the moment values are
obtained by solving the moment equations. Since the stochastic differential
equation(SDE) is needed to obtain the moment equations, the Autoregressive
Moving Average(ARMA) filter is used. The effective wave is modeled by using the
6th-order ARMA filter. In addition, the parametric excitation process is
modeled by using a non-memory transformation obtained from the relationship
between GM and wave elevation. The resulting system of equations is represented
by the 8th-order It\^o stochastic differential equation, which consists of a
second-order SDE for the ship motion and a 6th-order SDE for the effective
wave. This system has nonlinear components. Therefore, the cumulant neglect
closure method is used as higher-order moments need to be truncated.
Furthermore, the probability density function of roll angle is determined by
using moment values obtained from the SDE and the moment equation. Here, two
types of the probability density function are suggested and have a good
agreement.
