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In-Depth Information
The weak formulation (
9.26
) addresses the Heston model. Since the generator
H
MS
A
(
9.15
) has the same structure as the generator
A
(
9.16
) of the multi-scale
model of Example
9.1.1
,i.e.
n
v
=
1,
ξ(y)
=|
y
|
, we also obtain the well-posedness
of the weak formulation for this model.
We derive the bilinear form
a
SV
(
·
,
·
)
for the general diffusion SV model in (
9.11
),
(
9.18
) and give its weak formulation. The operator
A
+
A
η
−
r
in (
9.18
) can be
written as (compare with (
9.19
))
SV
f )(z)
(
A
:=
(
A
f
+
A
η
f
−
rf )(z)
1
2
tr
(z)D
2
f(z)
μ(z)
∇
:=
[
Q
]+
f(z)
+
c(z)f (z),
(9.27)
1
1
[
Q
D
2
η
]+
2
b)
∇
η
−
r
. Denote by
G
:=
where
μ
:=
Σ
η
+
b
, and
c
:=
2
tr
2
(Σ
η
+
G
Y
(
ln
(S), Y
1
,...,Y
n
v
)
. Then, the bilinear
R×
=
the state space of the process
Z
SV
form associated to the operator
A
:=
−
A
SV
ϕ,φ
,ϕ,φ
a
SV
(ϕ, φ)
C
0
∈
(G),
becomes, after integration by parts,
1
2
1
2
a
SV
(ϕ, φ)
ϕ)
Q
∇
2
μ)
∇
:=
(
∇
φ
d
x
+
(
∇
Q
−
ϕφ
d
x
−
cϕφ
d
x.
(9.28)
G
G
G
W
1
,
∞
(G)
(n
v
+
1
)
×
(n
v
+
1
)
L
∞
(G)
n
v
+
1
,
A
Herewith, we denote by
A
,
the operator which takes the divergence of the columns
A
k
:=
(A
1
,k
,...,A
n
v
+
1
,k
)
,
k
∇ :[
]
→[
]
→ ∇
=
1
,...,n
v
+
1, of the matrix
A
,i.e.
:=
div
(
A
1
),...,
div
(
A
n
v
+
1
)
.
Note that for constant coefficients
∇
A
,
μ
and
c
, the bilinear form
a
SV
(
Q
·
,
·
)
reduces to
the bilinear form
a
BS
(
)
in (8.9) of the multivariate Black-Scholes model.
The variational formulation to (
9.18
) reads:
Find
w
·
,
·
L
2
(J
H
1
(J
L
2
(G))
such that
∈
;
V)
∩
;
a
SV
(w, v)
(9.29)
+
=
∀
∈
(∂
t
w,v)
0
,
v
V,
a.e. in
J,
w(
0
)
=
w
0
.
(G)
·
V
, where the
C
0
Here, we assume that the Hilbert space
V
is given by
V
:=
·
V
depends on the model. We also assume that
a
SV
(
norm
·
,
·
)
:
V
×
V
→ R
is
L
2
(G)
.
continuous and satisfies a Gårding inequality, and that
w
0
∈
9.4 Localization
We consider the pricing equation (
9.18
) truncated to a bounded domain
G
R
:=
n
v
+
1
k
(a
k
,b
k
)
,
b
k
>a
k
∈ R
,
=
1
SV
w
R
+
∂
t
w
R
−
A
rw
R
=
0in
J
×
G
R
,
(9.30)
=
w
R
(
0
,z)
w
0
in
G
R
.
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