Volume 9, Issue 1

The aim of the paper is to give the proof of the Central Limit Theorem for a class of ( n, k )-particle processes solving balance equations for measures.

The paper is devoted to the stochastic solution of non-homogeneous Boltzmann-like equations up to ­ fixed time t with the help of the free-motion Kac particle systems. Sufficient conditions for the propagation of chaos are represented mainly in terms of distributions of point processes which govern jumps in Kac n -particle systems.

The Kusuoka approximation is a new simulation scheme for diffusion processes which are solutions of SDE with smooth coefficients. The author had reported that the Kusuoka approximation realizes several thousands times faster calculation of some financial derivative pricing problems than the Euler-Maruyama approximation does. In this paper, the author applied TBBA to the Kusuoka approximation and succeeded in several hundreds times faster calculation than naive Monte Carlo sampling.

Stochastic approximation is a method to search for the unique solution of the equation M( x ) = α under noisy observation. In this note, we are concerned with a discrete version of the method and its applications to data analysis.

The global algorithm of Random Walk on Spheres suggested in [Sabelfeld K.K. Monte Carlo methods in boundary problems. Springer-Verlag, Heidelberg - Berlin - New York, 1991.] is analyzed and a kind of optimization strategy is suggested. The algorithm is applied here to construct a functional version of this method which uses a multilinear interpolation. As an example we have chosen the biharmonic equation governing the bending of a thin elastic plate with the simply supported boundary, however generalizations to other equations can be carried out.

A sampling problem with incomplete data is considered: modelling of a random vector when only marginal distributions of its components and their correlation coefficients are known. Simplest modelling algorithms for such problems include repeated use of random values, and the realized joint distributions are singular. An alternative method without singularities is proposed.

We deal with evaluation of an integral with the positive integrand g ( v ) : V → R (here is a bounded domain) by standard Monte Carlo method with density p ( v ), proportional to the Strang–Fix approximation of function g ( v ), built on the uniform grid with step h in domain V . We consider the mean value of time required for modelling sampling value of random variable with respect to the density p ( v ) and investigate its asymptotic behaviour as h → 0.