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1
)
R
1
2
σ
2
1
2
σ
2
(ρ
+
μ)(
2
ρ
+
s
ρ
+
μ
ϕ
s
2
s
2
ρ
+
2
μ
−
2
ϕ
2
d
s
=
L
2
(G)
−
2
μ
−
0
2
μ)
R
1
2
r(
1
s
2
μ
ϕ
2
d
s
s
μ
ϕ
2
+
+
+
r
L
2
(G)
.
0
Given 1
/
2
≤
ρ<
1, we now choose
μ
such that
−
1
/
2
≤
μ<
1
/
2
−
ρ
. Then, 2
ρ
+
2
μ
−
1
<
0, 1
+
2
μ
≥
0,
ρ
+
μ
≥
0, and we get
1
2
σ
2
1
2
min
a
CEV
s
ρ
+
μ
∂
s
ϕ
2
s
μ
ϕ
2
σ
2
,
2
r
2
ρ,μ
(ϕ, ϕ)
≥
L
2
(G)
+
r
L
2
(G)
≥
{
}
ϕ
ρ,μ
.
By density of
C
0
(
0
,R)
in
W
ρ,μ
,wehaveshown(
4.32
).
We are now ready to cast Eq. (
4.19
) into the abstract parabolic framework. We
choose
μ
as in (
4.30
) and observe that
μ
s
2
μ
d
s)
denotes the weighted
L
2
-space corresponding to
W
ρ,μ
, we have the dense inclusions
L
2
(
0
,R
≤
0 then. Hence, if
H
μ
:=
;
→
H
μ
=
H
μ
)
∗
(W
ρ,μ
)
∗
.
W
ρ,μ
(
→
(4.33)
Denote by
(
·
,
·
)
μ
the inner product in
H
μ
. The weak formulation then reads:
Find
v
∈
L
2
(J
;
W
ρ,μ
)
∩
H
1
(J
;
H
μ
)
such that
a
CEV
(∂
t
v,w)
μ
+
ρ,μ
(v, w)
=
0
,
∀
w
∈
W
ρ,μ
,
a.e. in
J,
(4.34)
v(
0
)
=
g.
Applying Theorem 3.2.2,wehaveshown
Theorem 4.5.4
Let ρ
,
and assume that μ satisfies
(
4.30
).
Then
,
the prob-
lem
(
4.34
)
admits a unique solution
.
∈[
0
,
1
]
4.5.2 Local Volatility Models
We replace the constant volatility
σ
in the Black-Scholes model (1.1) by a deter-
ministic function
σ(s)
, i.e. the Black-Scholes model is extended to
d
S
t
=
rS
t
d
t
+
σ(S
t
)S
t
d
W
t
,
(4.35)
with
r
sσ(s)
satisfies
(1.3)-(1.4), such that the SDE (
4.35
) admits a unique solution. Thus, the infinitesi-
mal generator
∈ R
≥
0
and
σ
: R
+
→ R
+
. We assume that the function
s
→
A
of the process
S
is
1
2
s
2
σ
2
(s)∂
ss
f(s)
(
A
f )(s)
=
+
rs∂
s
f(s).
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