Triangular stochastic differential equations with boundary conditions
type: Item
instance of: scholarly article
title: Triangular stochastic differential equations with boundary conditions
zbMATH Open document ID: 0795.60051
author: Marco Ferrante
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1994-03-07T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108300
review text: The author considers the general equation \[ {d \over dt} X_ t + f(X_ t)={d \over dt} W_ t; \quad h(X_ 0,X_ 1)=0, \] where \(t \in [0,1]\), \(\{W_ t:t \in [0,1]\}\) is a \(d\)-dimensional Brownian motion, \(\{X_ t\}\) takes values in \(\mathbb{R}^ d\) and (note) the boundary condition involves both \(X_ 0\) and \(X_ 1\). It is also assumed that for each \(i \in \{1,\dots,d\}\), \(f_ i (x_ 1, \dots,x_ d)\) depends only on the first \(i\) variables, with a similar condition being imposed upon the \(h_ i\). A necessary condition is provided for the solution \(X_ t\) to be a Markov field. Then two ``necessary and sufficient'' results are given: in the first of these, \(X_ t\) is shown to be a Markov field process if and only if the \(f_ i\) are linear in the last variable, while in the second \(X_ t\) is a Markov field process if and only if the boundary condition is of a specific form.
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 510104
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: Markov property
zbMATH Keywords: Brownian motion
zbMATH Keywords: Markov field
zbMATH Keywords: boundary condition
reviewed by: Andrew Dale
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 510104