Volume 21, Issue 4

An algorithm is proposed for computing equilibrium averages of Markov chains which suffer from metastability – the tendency to remain in one or more subsets of state space for long time intervals. The algorithm, called the parallel replica method (or ParRep), uses many parallel processors to explore these subsets more efficiently. Numerical simulations on a simple model demonstrate consistency of the method. A proof of consistency is given in an idealized setting. The parallel replica method can be considered a generalization of A. F. Voter's parallel replica dynamics, originally developed to efficiently simulate metastable Langevin stochastic dynamics.

Maximum cross-section technique is used for solving problems of radiation transport by the Monte Carlo method to optimize the particles' free-path length modeling in inhomogeneous media. A probabilistic proof of a variation of this technique is proposed in the article.

By using main properties of uniformly distributed sequences of increasing finite sets in infinite-dimensional rectangles in ℝ ∞ described in [Real Anal. Exchange 36 (2010/2011), no. 2, 325–340], a new approach for an infinite-dimensional Monte-Carlo integration is introduced and the validity of some infinite-dimensional strong law type theorems are obtained in this paper. In addition, by using properties of uniformly distributed sequences in the unit interval, a new proof of Kolmogorov's strong law of large numbers is obtained which essentially differs from its original proof.

The standard Coupling From The Past (CFTP) algorithm is an interesting tool to sample from exact stationary distribution of a Markov chain. But it is very expensive in time consuming for large chains. There is a monotone version of CFTP, called MCFTP, that is less time consuming for monotone chains. In this work, we propose two techniques to get monotone chain allowing use of MCFTP: widening technique based on adding two fictitious states and clustering technique based on partitioning the state space in clusters. Usefulness and efficiency of our approaches are showed through a sample of Markov Chain Monte Carlo simulations.

We study the two-factor CIR model and using the main idea of two of our previous papers, we propose explicit numerical schemes that preserves positivity and converges strongly in the mean square sense to the true solution. Our results apply for the multi-factor and the one-factor cases as well.

Five algorithms for the simulation of random vectors with the bivariate Morgenstern distribution are described and realized. The run-time efficiencies of these algorithms are estimated so that the fastest one is determined. It uses the presentation of the Morgenstern distribution density as the sum of the bilinear finite elements.