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Zbl 1064.34068
Appleby, J.A.D.; Kelly, C.
Asymptotic and oscillatory properties of linear stochastic delay differential equations with vanishing delay. (English)
[J] Funct. Differ. Equ. 11, No. 3-4, 235-265 (2004). ISSN 0793-1786

The authors consider the scalar linear stochastic differential equation $$ dX(t)=(aX(t)+b(X(t-\tau(t))))\,dt+\sigma X(t)\,dB(t), \quad t\ge 0,$$ where the time lag $\tau$ is a continuous function vanishing at infinity and where $B$ is a standard Brownian motion. Depending on the decay of $\tau(t)$ to zero as $t\to\infty$, the solution process is proved to be almost surely oscillatory or nonoscillatory. The key ingredients for the proof are a random functional-differential equation solved by $X$ with a geometric Brownian motion as coefficient and related results for deterministic functional-differential equations. In addition, the long-time asymptotics of the solutions are studied in detail.
[Markus Reiss (Berlin)]

MSC 2000:: *34K50 Stochastic delay equations
34K11 Oscillation theory of functional-differential equations
60H10 Stochastic ordinary differential equations

Keywords: stochastic delay differential equation; oscillations; Lyapunov exponent

Cited in: Zbl 1122.65011

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