Conditioning a reflected one-dimensional diffusion via its canonical decomposition
type: Item
instance of: scholarly article
title: Conditioning a reflected one-dimensional diffusion via its canonical decomposition
zbMATH Open document ID: 0773.60074
DOI: 10.1007/BF01047576
published in: Journal of Theoretical Probability
publication date: 1993-10-25T00:00:00Z
review text: The author considers a reflected one-dimensional diffusion process \(X\) starting from \(x\geq 0\), the point 0 being an instantaneously reflecting boundary. If the scale function is convex and if appropriate conditions on the speed measure are fulfilled, the diffusion \(X\) can be written as \(X=N-B\), where \(B\) is a Brownian motion starting from \(-x\) and \(X\) a locally zero-energy additive functional. The author studies the diffusion conditioned by the event ``\(N\leq 0\)''. The conditional law \(P(\cdot/N\leq 0)\) is obtained as the limit of the conditional laws \(P(\cdot/vert N\leq y)\) when \(y\) goes to zero. The proof of this result uses a path decomposition analogous to the one of D. Williams. Then, the author gives a simple pathwise construction of the conditioned process from the original diffusion \(X\).
reviewed by: Mireille Chaleyat-Maurel
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60J50
zbMATH DE Number: 179066
zbMATH Keywords: one-dimensional diffusion process
zbMATH Keywords: Brownian motion
zbMATH Keywords: zero-energy additive functional
zbMATH Keywords: path decomposition
zbMATH Keywords: pathwise construction
author: Jean Bertoin
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01047576
OpenAlex ID: W2090590568
description: scientific article; zbMATH DE number 179066