On the Markov property of a stochastic difference equation
type: Item
instance of: scholarly article
title: On the Markov property of a stochastic difference equation
zbMATH Open document ID: 0811.60050
DOI: 10.1016/0304-4149(94)90027-2
author: Marco Ferrante
author: David Nualart
published in: Stochastic Processes and their Applications
publication date: 1995-04-19T00:00:00Z
review text: Let \(\{X_ n\), \(0\leq n\leq N-1\}\), \(N>6\), be a discrete time process which satisfies the two-point boundary condition stochastic difference system \[ X_{n+1}= X_ n+ f(X_ n)+ \sigma(X_ n)\xi_ n, \;0\leq n\leq N-1, \qquad F_ 0 X_ 0+ X_ N=F, \tag \(*\) \] where \(\{\xi_ i\), \(0\leq i\leq N-1\}\) are independent variables, \(f\in C^ 2\), \(\sigma\in C^ 3\) with \(f(0)= 0=\sigma(0)\), \(f'(x)\geq -1\), \(\sigma'(x)>0\) for all \(x\) in \((0,\infty)\), the \(\xi_ n\) are positive, and \(F_ 0>0\), \(F>0\). The authors show that \((*)\) has a unique solution and give necessary and sufficient conditions that \((*)\) be a reciprocal Markov chain. [The authors call a sequence of random variables \(\{X_ 0,\dots, X_ M\}\) a reciprocal Markov chain if for every \(m\) and \(n\) with \(0\leq m< n-1<M-1\), the \(\sigma\)-fields \(\sigma(X_{m+1},\dots, X_ n)\) and \(\sigma(X_ 0,\dots, X_ m, X_{n+1},\dots, X_ M)\) are conditionally independent given \(X_ n\) and \(X_ m\)].
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 60J10
zbMATH DE Number: 691007
zbMATH Keywords: two-point boundary condition stochastic difference system
zbMATH Keywords: reciprocal Markov chain
zbMATH Keywords: conditionally independent
reviewed by: Kenneth S. Miller
MaRDI profile type: Publication
full work available at URL: 0304-4149(94)90027-2
OpenAlex ID: W2083520118
cites work: Q3142407
cites work: Equations différentielles stochastiques dans avec conditions aux bords
cites work: Q3038308
cites work: Stochastic calculus with anticipating integrands
cites work: Boundary value problems for stochastic differential equations
description: scientific article; zbMATH DE number 691007