Affiliation:
1. Department of Mathematics , Faculty of Education-Ateq , Shabwh University , Shabwah , Yemen
2. Department of Mathematics , Faculty of Science , Aden University , Aden , Yemen
Abstract
Abstract
In this paper, under some suitable conditions, we prove existence of a strong solution and uniqueness for the perturbed stochastic differential equations with reflected boundary (PSDERB), that is,
{
x
(
t
)
=
x
(
0
)
+
∫
0
t
σ
(
s
,
x
(
s
)
)
d
B
(
s
)
+
∫
0
t
b
(
s
,
x
(
s
)
)
d
s
+
α
(
t
)
H
(
max
0
≤
u
≤
t
x
(
u
)
)
+
β
(
t
)
L
t
0
(
x
)
,
x
(
t
)
≥
0
for all
t
≥
0
,
\left\{\begin{aligned} {}x(t)&=x(0)+\int_{0}^{t}\sigma(s,x(s))\,dB(s)+\int_{0}^{t}b(s,x(s))\,ds+\alpha(t)H\bigl{(}\max_{0\leq u\leq t}x(u)\bigr{)}+\beta(t)L_{t}^{0}(x),\\
x(t)&\geq 0\quad\text{for all}\ t\geq 0,\end{aligned}\right.
where 𝐻 is a continuous R-valued function,
σ
,
b
,
α
\sigma,b,\alpha
and 𝛽 are measurable functions,
L
t
0
L_{t}^{0}
denotes a local time at point zero for the time of the semi-martingale 𝑥.
Subject
Applied Mathematics,Statistics and Probability
Cited by
1 articles.
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