Volume 16, Issue 2

Approximate formulas for evaluation of mathematical expectations of nonlinear functionals of solutions to Ito's stochastic differential equation are constructed. The general approach is based on quadrature formulas which are exact for functional polynomials.

We present a method for numerical modeling of binary homogeneous random fields based on thresholds of Gaussian functions. The method enables us to reproduce the average value and correlation function of the observed binary field. The method is comparatively simple, and on several examples we demonstrate that it can be effective for simulation of a wide class of binary random fields.

A Monte Carlo algorithm is developed for generating samples of arbitrary real-valued nonstationary Gaussian processes. The algorithm is based on representations of these processes by finite sums of harmonics with dependent Gaussian coefficients, in contract to similar representations available for stationary Gaussian processes that have independent Gaussian coefficients. The proposed algorithm is based on an idea in [Grigoriu, J. Eng. Mech., ASCE 119: 328–343, 1993]. It is shown that the covariance matrix of the random coefficients in the representation proposed for nonstationary Gaussian processes can be obtained simply from ordinates of the generalized spectral densities of these processes. Two numerical examples are presented to illustrate the application of the proposed algorithm and assess its performance. The examples are a stationary Gaussian process and a nonstationary Gaussian process obtained from a stationary process by distorting its time scale.

In this paper the Multilevel Monte Carlo method introduced by Giles [Operations Research 56: 607–617, 2008] is refined by the extended complexity theorem and applied to Lévy process driven SDEs. The restricted approximate Euler scheme is used to simulate the SDEs. Recent results about the convergence by Fournier [ArXiv e-prints: 2009] and by Jacod, Kurtz, Máléard and Protter [Annales de l'Institut Henri Poincare (B) Probability and Statistics 41: 523–558, 2005] are used. The theoretical results are illustrated by numerical simulations.

Erratum for the article “Exact retrospective Monte Carlo computation of arithmetic average Asian options” which appeared in Monte Carlo Methods Appl. 13 (2007), 135–171.