THE SHIFT: Properties and recommendations for practical use

In mechanics of structures as well as in other domains of engineering, the probabilistic models often have to be computed by simulation. One of the reasons is that the stochastic calculus involved in non linear representations yields rarely explicit formulas. Usually these simulations use, for each sample, a large number of calls to the random function, in this case the simulation by the shift, whose field of application is as wide as that of Monte Carlo method, is particularly relevant. The theoretical features, the implementation and the specific advantages of this method have been taught in a course of the author at Paris VI University in 1988 and are detailed in the book with D. Lépingle [BL]. I. Other known methods of simulation Let us begin by the presentation of the methods to be compared with the shift. A. The Monte Carlo method For computing the expectation IE(X) of a random variable X, it consists of a) representing X as a random variable defined on the probability space ([0, 1],B[0, 1], dx1dx2 · · · dxs) with s finite or infinite, b) imitating independent random samples of points of [0, 1], (U11, U12, . . . , U1s) , (U21, U22, . . . , U2s) , . . . , (Un1, Un2, . . . , Uns) , . . . c) and applying the law of large numbers IE(X) = lim N↑∞ 1 N N