Martingale Approach to Monte Carlo Simulation and Linear Random Vibration
Publication: Journal of Engineering Mechanics
Volume 125, Issue 12
Abstract
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.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Chung, K. L. (1974). A course in probability theory. Academic Press, New York.
2.
Chung, K. L., and Williams, R. J. (1990). Introduction to stochastic integration, 2nd Ed., Birkhäuser, Boston.
3.
Grigoriu, M. (1992). “Transient response of linear systems to stationary Gaussian inputs.” Probabilistic Engrg. Mech., 7, 159–164.
4.
Grigoriu, M. (1995). Applied non-Gaussian processes: Examples, theory, simulation, linear random vibration, and MATLAB solutions. Prentice-Hall, Englewood Cliffs, N.J.
5.
Krishnan, V. (1984). Nonlinear filtering and smoothing: An introduction to martingale, stochastic integrals and estimation. Wiley, New York.
6.
Protter, P. (1990). Stochastic integration and differential equations. Springer, New York.
7.
Soong, T. T., and Grigoriu, M. (1992). Random vibration of mechanical and structural systems. Prentice-Hall, Englewood Cliffs, N.J.
Information & Authors
Information
Published In
History
Received: Nov 25, 1998
Published online: Dec 1, 1999
Published in print: Dec 1999
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.