Conditional Monte Carlo method for dynamic systems with random properties

A method is developed for calculating moments and other properties of states X(t) of dynamic systems with random coefficients depending on semi-Markov processes ξ(t) and subjected to Gaussian white noise. Random vibration theory is used to find probability laws of conditional processes X(t)∣ξ(·). Unconditional properties of X(t) are estimated by averaging conditional statistics of this process corresponding to samples of ξ(t). The method is particularly efficient for linear systems since X(t)∣ξ(·) is Gaussian during periods of constant values of ξ(t), so that and its probability law is completely defined by the process mean and covariance functions that can be obtained simply from equations of linear random vibration. The method is applied to find statistics of an Ornstein-Uhlenbeck process X(t) whose decay parameter is a semi-Markov process ξ(t). Numerical results show that X(t) is not Gaussian and that the law of this process depends essentially on features of ξ(t). A version of the method is used to calculate the failure probability for an oscillator with degrading stiffness subjected to Gaussian white noise.