Weak existence of the squared Bessel and CIR processes with skew reflection on a deterministic time-dependent curve
Let σ > 0 , δ ≥ 1 , b ≥ 0 , 0 < p < 1 . Let λ be a continuous and positive function in H l o c 1 , 2 ( R + ) . Using the technique of moving domains (see Russo and Trutnau (2005) [9]), and classical direct stochastic calculus, we construct for positive initial conditions a pair of continuo...
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Published in | Stochastic processes and their applications Vol. 120; no. 4; pp. 381 - 402 |
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Main Author | |
Format | Journal Article |
Language | English |
Published |
Amsterdam
Elsevier B.V
01.04.2010
Elsevier |
Series | Stochastic Processes and their Applications |
Subjects | |
Online Access | Get full text |
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Summary: | Let
σ
>
0
,
δ
≥
1
,
b
≥
0
,
0
<
p
<
1
. Let
λ
be a continuous and positive function in
H
l
o
c
1
,
2
(
R
+
)
. Using the technique of moving domains (see Russo and Trutnau (2005)
[9]), and classical direct stochastic calculus, we construct for positive initial conditions a pair of continuous and positive semimartingales
(
R
,
R
)
with
d
R
t
=
σ
R
t
d
W
t
+
σ
2
4
(
δ
−
b
R
t
)
d
t
+
(
2
p
−
1
)
d
ℓ
t
0
(
R
−
λ
2
)
,
and
d
R
t
=
σ
2
d
W
t
+
σ
2
8
(
δ
−
1
R
t
−
b
R
t
)
d
t
+
(
2
p
−
1
)
d
ℓ
t
0
(
R
−
λ
)
+
I
{
δ
=
1
}
2
d
ℓ
t
0
+
(
R
)
,
where the symmetric local times
ℓ
0
(
R
−
λ
2
)
,
ℓ
0
(
R
−
λ
)
, of the respective semimartingales
R
−
λ
2
,
R
−
λ
are related through the formula
2
R
d
ℓ
0
(
R
−
λ
)
=
d
ℓ
0
(
R
−
λ
2
)
.
Well-known special cases are the (squared) Bessel processes (choose
σ
=
2
,
b
=
0
, and
λ
2
≡
0
, or equivalently
p
=
1
2
), and the Cox–Ingersoll–Ross process (i.e.
R
, with
λ
2
≡
0
, or equivalently
p
=
1
2
). The case
0
<
δ
<
1
can also be handled, but is different. If
|
p
|
>
1
, then there is no solution. |
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ISSN: | 0304-4149 1879-209X |
DOI: | 10.1016/j.spa.2010.01.005 |