Explosion time of second-order Ito processes
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 \(F(u)=\int^{u}_{0}f(s)ds\).
scientific article; zbMATH DE number 3874355