Martingale Approach to Monte Carlo Simulation and Linear Random Vibration

A method is developed for generating samples of the state X of a linear filter driven by Gaussian white noise. The method can also be applied to developed formulas giving the second moment properties of X. The proposed solution is based on two facts. First, the integrals defining the forcing component of X are continuous martingales because their integrands are continuous deterministic functions and their integrators are Brownian motions. Second, a continuous martingale satisfying some additional properties can be time changed to a standard Brownian motion. The proposed Monte Carlo algorithm for generating samples of X calculates samples of this process from samples of a standard Brownian motion defined in a new clock by mapping these samples to the original clock. The algorithm does not use recurrence formulas for generating samples of X and does not approximate the probability law of this process. Examples are presented to illustrate the application of the proposed method. The examples include the Ornstein-Uhlenbeck process and the response of a simple oscillator and multi-degree-of-freedom system to Gaussian white noise.