Applications of some formulas for finite Markov chains
type: Item
instance of: scholarly article
title: Applications of some formulas for finite Markov chains
zbMATH Open document ID: 1211.60033
author: Maxim J. Goldberg
author: Seonja Kim
published in: Applied and Computational Harmonic Analysis
publication date: 2011-01-13T00:00:00Z
review text: The authors describe how some elementary computations with finite state Markov chain transition matrices lead to interesting and promising applications. The computations and sample applications are based on two observations. The first one is that the product of the transition matrix interlaced with projections on states of interest gives probabilities along corresponding subsets of paths. The scond observation is a discrete version of the Feynman-Kac formula. Finally four applications are presented. In one of these applications the two observations are combined to yield an effective computational method for evaluating the expected value (over paths) of a function of the sum of path attributes; each path starts at a specific point within a subset of states, and ends when the path exits the given subset for the first time. At the same time they describe some sample problems for which their methods may be of use. The authors propose, in another application, a variant of the occasionally used approach to calculate solutions to a large class of Dirichlet-Poisson boundary value problems via diffusion. The proposed method finds the exact solution for each discretization of the continuous boundary value problem. It is emphasized that the existing stochastic methods of which they are aware rely on Monte Carlo simulations and thus require a large number of trials to achieve good accuracy.
reviewed by: Gabriel V. Orman
Mathematics Subject Classification ID: 60J22
Mathematics Subject Classification ID: 60J27
zbMATH DE Number: 5835729
zbMATH Keywords: finite Markov chains
zbMATH Keywords: Feynman-Kac formula
zbMATH Keywords: pmf of costs along paths
zbMATH Keywords: boundary value problems
zbMATH Keywords: exact expected values for random walks
MaRDI profile type: Publication
full work available at URL: j.acha.2010.02.004
OpenAlex ID: W2020886575
cites work: Diffusion wavelets
cites work: On Occupation Times for Markoff Processes
cites work: Space-Time Approach to Non-Relativistic Quantum Mechanics
cites work: Kac's moment formula and the Feynman-Kac formula for additive functionals of a Markov process
cites work: Integration in Functional Spaces and its Applications in Quantum Physics
cites work: Q4003764
cites work: On the moments of Markov renewal processes
cites work: Generalized inverses and their application to applied probability problems
cites work: Q2779467
cites work: A Feynman-Kac path-integral implementation for Poisson's equation using an \(h\)-conditioned Green's function
cites work: On Distributions of Certain Wiener Functionals
cites work: Q5808624
cites work: Q3262596
cites work: Solving boundary value problems with complex parameters by the Monte Carlo method
cites work: The Simplest Random Walks for the Dirichlet Problem
cites work: Numerical methods for linear boundary value problems based on Feynman-Kac representations
cites work: Stochastic methods for Dirichlet problems
DOI: 10.1016/J.ACHA.2010.02.004
description: scientific article; zbMATH DE number 5835729