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Using [143, Theorem 25.18], it suffices to show that there exist a constant
C(T,σ) >
0 such that
e
q
|
X
T
|
1
x
C(T,σ)e
−
γ
1
R
+
γ
2
|
x
|
.
E
}
|
X
t
=
≤
{|
X
T
|
>R
σ
2
/
2, with the transition probability
p
T
−
t
,
We have for
μ
=
r
−
e
q
|
X
T
|
1
}
|
X
t
=
x
e
q
|
z
+
x
|
1
E
=
p
T
−
t
(z)
d
z
{|
X
T
|
>R
{|
z
+
x
|
>R
}
R
e
q
|
x
|
1
2
πσ
2
(T
e
−
(z
−
μ(T
−
t))
2
/(
2
σ
2
(T
−
t))
d
z
e
q
|
z
|
1
{|
z
+
x
|
>R
}
≤
−
t)
R
C
1
(T , σ ) e
q
|
x
|
e
(q
+
μ/σ
2
)
|
z
|
1
{|
z
+
x
|
>R
}
e
−
z
2
/(
2
σ
2
(T
−
t))
d
z
≤
R
≤
C
1
(T , σ ) e
q
|
x
|
e
−
(η
−
q
−
μ/σ
2
)(R
−|
x
|
)
e
η
|
z
|
e
−
z
2
/(
2
σ
2
(T
−
t))
d
z
R
C
1
(T , σ ) e
−
γ
1
R
+
γ
2
|
x
|
e
η
|
z
|
e
−
z
2
/(
2
σ
2
(T
−
t))
d
z,
≤
R
q
. Since
R
e
η
|
z
|
e
−
z
2
/(
2
σ
2
(T
−
t))
d
z<
μ/σ
2
, and
γ
2
=
with
γ
1
=
η
−
q
−
γ
1
+
∞
for
μ/σ
2
.
any
η>
0, we obtain the required result by choosing
η>q
+
Remark 4.3.2
We see from Theorem
4.3.1
that
v
R
→
v
exponentially for a fixed
x
→∞
=±
as
R
. The artificial zero Dirichlet barrier type conditions at
x
R
are
not
describing correctly the asymptotic behavior of the price
v(t,x)
for large
|
x
|
. Since
the barrier option price
v
R
is a good approximation to
v
for
|
x
|
R
,
R
should be
selected substantially larger than the values of
x
of interest.
The barrier option price
v
R
can again be computed as the solution of a PDE
provided some smoothness assumptions.
C
1
,
2
(J
C
0
(J
Theorem 4.3.3
Let v
R
(t, x)
∈
× R
)
∩
× R
) beasolutionof
BS
v
R
−
∂
t
v
R
+
A
rv
R
=
0
(4.12)
on (
0
,T)
×
G where the terminal and boundary conditions are given by
g(e
x
),
G
c
.
v
R
(T , x)
=
∀
x
∈
G,
v
R
(t, x)
=
0
on (
0
,T)
×
Then
,
v
R
(t, x) can also be represented as in
(
4.11
).
Now we can restate the problem (
4.8
) on the bounded domain:
L
2
(J
H
0
(G))
H
1
(J
L
2
(G))
such that
Find
u
R
∈
;
∩
;
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