The Picard boundary value problem for a third order stochastic difference equation
type: Item
instance of: scholarly article
title: The Picard boundary value problem for a third order stochastic difference equation
zbMATH Open document ID: 0874.60056
author: Marco Ferrante
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1997-11-02T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108416
review text: Summary: It is considered the multidimensional third-order stochastic difference equation \[ \Delta^3X_{n-2}= f(X_n)+ \xi_n,\quad n\in\{2,\dots,N-1\},\quad N\geq 5, \] where \(X_i\in\mathbb{R}^d\), \(d\geq 1\), and \(\{\xi_i\}\) is a sequence of \(d\)-dimensional independent random vectors, with the Picard boundary condition \[ X_0= a_0,\quad X_1=a_1,\quad X_N= a_N,\quad a_i\in\mathbb{R}^d,\quad i=0,1,N. \] We first prove that the boundary value problem admits a unique solution if \(f\) is a monotone application. Moreover, we are able to compute the density of the law of the solution if the random vectors \(\{\xi_i\}\) are absolutely continuous. Thanks to this explicit computation, in the scalar case we prove that the process \(\{(X_i,\Delta X_i,\Delta^2X_i): i=0,\dots,N-2\}\) is a Markov chain if and only if \(f\) is affine and we provide a simple counterexample to show that a similar strong condition does not hold in the multidimensional case.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 1021761
zbMATH Keywords: stochastic difference equation
zbMATH Keywords: Picard boundary condition
zbMATH Keywords: boundary value problem
MaRDI profile type: Publication
cites work: Q3142407
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description: scientific article; zbMATH DE number 1021761