Probabilistic approach to large amplitude ship rolling in random seas

For a vessel in open seas, the sudden indirect excitation of roll motions due to waves from the front or rear leads to dangerous situations, sometimes even capsizing. We derive a general non-linear model, which is appropriate for the analysis of parametric excited roll motions in head or following random seas. The irregular waves are modelled in terms of a continuous time autoregressive moving average process. The resulting model of stochastic differential equations is investigated numerically by Local Statistical Linearization. The necessary stochastic moments and their derivatives are computed using Itô's differential rule and Gaussian closure.

Representations of continuous-time ARMA processes

Using the kernel representation of a continuous-time Lévy-driven ARMA (autoregressive moving average) process, we extend the class of nonnegative Lévy-driven Ornstein–Uhlenbeck processes employed by Barndorff-Nielsen and Shephard (2001) to allow for nonmonotone autocovariance functions. We also consider a class of fractionally integrated Lévy-driven continuous-time ARMA processes obtained by a simple modification of the kernel of the continuous-time ARMA process. Asymptotic properties of the kernel and of the autocovariance function are derived.

Representations of continuous-time ARMA processes

Using the kernel representation of a continuous-time Lévy-driven ARMA (autoregressive moving average) process, we extend the class of nonnegative Lévy-driven Ornstein–Uhlenbeck processes employed by Barndorff-Nielsen and Shephard (2001) to allow for nonmonotone autocovariance functions. We also consider a class of fractionally integrated Lévy-driven continuous-time ARMA processes obtained by a simple modification of the kernel of the continuous-time ARMA process. Asymptotic properties of the kernel and of the autocovariance function are derived.