DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

ASYMPTOTIC BEHAVIOR IN STOCHASTIC FUNCTIONAL
DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE
ZEPHYRINUS C. OKONKWO
This paper deals with asymptotic behavior of the solution process of a class of neutral
stochastic functional differential equations of Itô-Volterra form. Criteria for the existence
of the solution process are outlined. Using the results of Corduneanu, Mahdavi,
and Okonkwo, asymptotic behaviors of such solution processes (at +∞) are discussed.
Copyright © 2006 Zephyrinus C. Okonkwo. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
We study the asymptotic behavior of the solutions to stochastic functional differential
equations of the form
with the random initial condition
d(Vx)(t,ω) = (Ax)(t,ω)dt + φ ( t,x(t,ω) ) dz(t,ω) (1.1)
x(0,ω) = x 0 ∈ R n . (1.2)
Here and in the sequel, V and A will denote causal operators on the function space
C(R + × Ω,R n ), which is the space of product measurable random functions x : R + × Ω →
R n with continuous sample paths on every compact subset of R + . Ω isthesamplespace,
with ω ∈ Ω. φ ∈ M 0 (R + × R n ,R n×k ), the space of n × k product measurable matrix-valued
random functions with the property that
( ∫ T1
∣
ω : sup
φ ( t,x(t,ω) )∣ )
∣ 2 dt < ∞ = 1, T 1 < ∞. (1.3)
0 0≤t≤T 1
The integral in (1.3) is assumed to be in the Lebesgue sense for each ω ∈ Ω. Wewill
present a remark concerning the asymptotic behavior of (1.1), where the underlying space
is L p loc (R + × Ω,R n ), 1 ≤ p ≤∞.
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 895–903