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Zbl 1105.34057
Appleby, John A.D.; Buckwar, Evelyn
Noise induced oscillation in solutions of stochastic delay differential equations.
(English)
[J] Dyn. Syst. Appl. 14, No. 2, 175-196 (2005). ISSN 1056-2176

The authors consider the linear stochastic delay differential equation $$dX(t)= (aX(t)+bX(t-\tau(t)))\,dt +\sigma X(t)\,dB(t),$$ $$X(t)=\psi(t),\qquad -\overline\tau\le t\le0,$$ where $\tau(t)\le\overline\tau$ is a continuous function, and $\psi\in C[-\overline\tau,0]$. The main results of the paper are the following statements: (a) If $b<0$, $0<\underline{\tau}<\tau(t)\le\overline\tau<\infty$, and $t\mapsto t-\tau(t)$ is a nondecreasing function, then the equation has an a.s.\ oscillatory solution $X$ on $[0,\infty)$, i.e., $\sup Z_X=\infty$ a.s., where $Z_X:=\{t>0:X(t)=0\}$. Moreover, all points of the zero set $Z_X$ are isolated, and $X$ is differentiable at all these points. (b) If, otherwise, $b>0$ and $\psi>0$ on $[-\overline\tau,0]$, then the equation has an a.s.\ positive solution.
[Vigirdas Mackevičius (Vilnius)]
MSC 2000:
*34K50 Stochastic delay equations
60H10 Stochastic ordinary differential equations
34K11 Oscillation theory of functional-differential equations

Keywords: Stochastic delay equations; oscillation of solutions; zero set of a stochastic process

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