Volume 10, Issue 1

In this article we propose an adaptative variance reduction method for Monte Carlo simulations. The method uses importance sampling scheme based on a change of drift. The change of drift is selected adaptatively through the Monte Carlo computation by using a suitable sequence of approximation. We state and prove theoretical results supporting the use of the method. We develop two applications of the procedure for variance reduction in a Monte Carlo computation in finance and in reliability.

Optimal prediction approximates the average solution of a large system of ordinary differential equations by a smaller system. We present how optimal prediction can be applied to a typical problem in the field of molecular dynamics, in order to reduce the number of particles to be tracked in the computations. We consider a model problem, which describes a surface coating process, and show how asymptotic methods can be employed to approximate the high dimensional conditional expectations, which arise in optimal prediction. The thus derived smaller system is compared to the original system in terms of statistical quantities, such as diffusion constants. The comparison is carried out by Monte-Carlo simulations, and it is shown under which conditions optimal prediction yields a valid approximation to the original system.

In a first part of the paper a simulation method for a strictly stationary non-Gaussian process with given one-dimensional marginal distribution (or N -first statistical moments) and autocorrelation function is recalled. This method was already widely treated in the articles [14] and [13]. The objective of the present paper is twofold: first, to simplify this method - if by Mehler formula it is possible to find an autocorrelation function yielding the target autocorrelation function, and second, analyze the difference between the given autocorrelation function and the model one.

A generalized process , where E is a nuclear space, is a random variable family such that the map is linear and continuous. The Gaussian white noise process is a well-known example. It is characterized by a Gaussian measure on the dual space E ′ of E . Ordinary Gaussian processes can be constructed from the white noise process using the duality relation: where is a family of functions in E . The goal of this paper is to show that all the classical simulation methods of Gaussian processes found in the literature are derived from this construction, by fixing the appropriate nuclear space E and family .