Explosion time of second-order Ito processes
type: Item
instance of: scholarly article
title: Explosion time of second-order Ito processes
zbMATH Open document ID: 0548.60066
DOI: 10.1016/0022-247X(84)90005-2
author: N. E. Zubov
published in: Journal of Mathematical Analysis and Applications
publication date: 1984-01-01T00:00:00Z
review text: During the last several years a number of exceptionally sharp results regarding uniqueness, continuation, and oscillation of solutions of the nonlinear second-order equation ÿ\(+a(t)f(y)=0\) have been obtained. Our work is concerned with the harmonic oscillator ÿ\(+g(t,y,\dot y)\dot y+a(t)f(y)=h(t,y,\dot y)\dot w\), where \(\dot w\) is the so-called white noise. For the response of the oscillator we take the solution of the two-dimensional stochastic differential equation. A sufficient condition is given for the existence of the global solution as well as a sufficient condition for the non-existence of the global solution. Under a certain condition, we find that there exists a global solution for such a function f(y) as \(yf(y)>0\) with \(y\neq 0\) and that there exists a non-global solution for such a function f(y) as \(yf(y)>0\) with \(y\neq 0\), satisfying both \(\int^{\infty}_{0}(1+F(u))^{- 1/2}du<\infty\) and \(\int^{-\infty}_{0}(1+F(u))^{-1/2}du>-\infty\), where \(F(u)=\int^{u}_{0}f(s)ds\).
Mathematics Subject Classification ID: 60H20
zbMATH DE Number: 3874355
zbMATH Keywords: explosion time
zbMATH Keywords: harmonic oscillator
zbMATH Keywords: existence of the global solution
MaRDI profile type: Publication
cites work: On Continuability of Solutions of Second Order Differential Equations
cites work: Second Order Itô Processes
cites work: Q3959169
cites work: Q5585820
cites work: No explosion criteria for stochastic differential equations
description: scientific article; zbMATH DE number 3874355