IJPAM: Volume 74, No. 2 (2012)

REMARKS ON THE STABILISATION OF STOCHASTIC
DELAY DIFFERENTIAL EQUATIONS BY NOISE

Olufemi Adeyinka Adesina$^1$, Titilope Christianah Oshinubi$^2$,
Johnson Adekunle Osilagun$^3$, Suraju Olusegun Ajadi$^4$
$^{1,2,4}$Department of Mathematics
Obafemi Awolowo University
Ile-Ife, NIGERIA
$^3$Department of Mathematical Sciences
Olabisi Onabanjo University
Ago-Iwoye, NIGERIA


Abstract. This paper is concerned with the stabilisation of finite dimensional stochastic delay differential equation of the form

\begin{displaymath}x'(t) = f_{1}\big( t, x(t),x(t-\tau_{1}(t),...,x(t-\tau_{n}(t))) + \int^{t}_{t-\tau_{0}(t)} f_{2}(s,t,x(s))ds, \qquad t\geq 0,\end{displaymath}

where $n\in \mathbf{N}$, $f_1 \in C(\mathbf{R} ^ + \times (\mathbf{R} ^d )^{n + 1} ;\mathbf{R} ^d )$, $f_2 \in C(\mathbf{R} ^ + \times \mathbf{R} ^ + \times \mathbf{R} ^d ;\mathbf{R} ^d )$. Under more general conditions, we show that a linear multiplicative noise can always stabilise a general finite-dimensional functional differential equation whenever the delay is sufficiently small. Our results complement and improve existing results, and are also in line with the results of Appleby [1] which in itself generalised some well known results in the literature.

Received: June 16, 2011

AMS Subject Classification: 39A11, 60F15, 65C30, 34D40, 34D45

Key Words and Phrases: stochastic delay differential equations, Ito's theorem, stability, noise

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 74
Issue: 2