On oscillations of the geometric Brownian motion with time-delayed drift
type: Item
instance of: scholarly article
title: On oscillations of the geometric Brownian motion with time-delayed drift
zbMATH Open document ID: 1126.60048
author: Uwe Küchler
author: A. A. Gushchin
published in: Statistics \& Probability Letters
publication date: 2007-03-02T00:00:00Z
full work available at URL: http://edoc.hu-berlin.de/18452/4218
review text: The authors consider the Ito stochastic differential equation \[ dX(t)=(aX(t)+f(X(t-r)))dt+\sigma X(t)dW(t),\quad t\geq 0 \] with scalar Brownian motion \(W\) and a locally bounded measurable function \(f\). Expressing the solution \(X\) in terms of the classical geometric Brownian motion, it can be proved that for a positive initial segment \((X(s),-r\leq s\leq 0)\) and non-negative \(f\), the process \(X\) remains positive a.s. On the other hand, the authors establish a condition on \(a\), \(\sigma\) and \(f\) such that the solution process with positive initial condition attains zero in finite time a.s. This condition is for instance satisfied if \(f\) is non-increasing with at least linear growth while \(a\) and \(\sigma\) are arbitrary.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 34K50
Mathematics Subject Classification ID: 93E03
zbMATH DE Number: 5130370
zbMATH Keywords: geometric Brownian motion
zbMATH Keywords: stochastic delay differential equations
reviewed by: Markus Reiß
MaRDI profile type: Publication
OpenAlex ID: W2095396596
cites work: Q3369828
cites work: Complete Models with Stochastic Volatility
cites work: Q4002114
cites work: Q3759218
cites work: Q4369402
DOI: 10.1016/J.SPL.2004.03.013
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description: scientific article; zbMATH DE number 5130370