Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
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

Highlights
Master Server

### Zentralblatt MATH Berlin [Germany]

Zentralblatt MATH master server is maintained by the Editorial Office in Berlin, Section Mathematics and Computer Science of FIZ Karlsruhe and is updated daily.

Other Mirror Sites

Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences