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where
V
is given in (
9.23
). The variational formulation of (
9.35
) reads:
Find
w
L
2
(J
H
1
(J
L
2
(G))
such that
w(t,
∈
;
V)
∩
;
·
)
∈
K
0
and
a
κ
(w, v
f
κ
,v
(∂
t
w,v
−
w)
+
−
w)
≥
−
w
V
∗
,V
,
∀
v
∈
K
0
,
a.e. in
J,
=
w(
0
)
0
.
(9.36)
Since the bilinear form
a
κ
(
)
is continuous and satisfies a Gårding inequality in
V
by Theorem
9.3.1
, problem (
9.36
) admits a unique solution for every payoff
g
·
,
·
∈
L
∞
(
)
by Theorem B.2.2 of the Appendix B. We localize the problem to a bounded
domain as in Sect.
9.4
and obtain the following problem for
K
0
,R
:=
v
R
G
R
,
∈
V
|
v
≥
0a.e.
z
∈
namely
Find
w
L
2
(J
;
V)
H
1
(J
L
2
(G
R
))
such that
w(t,
∈
∩
;
·
)
∈
K
0
,R
and
a
H
(w, v
f
κ
,v
(∂
t
w,v
−
w)
+
−
w)
≥
−
w
V
∗
,V
,
∀
v
∈
K
0
,R
,
a.e. in
J,
w(
0
)
=
0
.
(9.37)
For a general diffusion SV model, the weak localized formulation reads:
Find
w
∈
L
2
(J
;
V)
∩
H
1
(J
;
L
2
(G
R
))
such that
w(t,
·
)
∈
K
0
,R
and
(∂
t
w,v
a
SV
(w, v
a
SV
(w
0
,v
−
w)
+
−
w)
≥−
−
w),
∀
v
∈
K
0
,R
,
a.e. in
J,
w(
0
)
=
0
,
(9.38)
K
0
,R
and
V
depend on the model. Discretization using finite differences or
finite elements in space and the backward Euler scheme in time as explained in
Sect.
9.5
leads to the following sequence of linear complementary problems for
(
9.37
). Given
w
0
N
=
where
0
, find
w
m
+
1
N
N
∈ R
such that for
m
=
0
,...,M
−
1,
B
w
m
+
1
N
F
m
,
≥
w
m
+
1
N
(9.39)
≥
0
,
)
B
w
m
+
1
N
F
m
=
(
w
m
+
1
N
−
0
,
M
u
N
f
κ
,b
i
V
∗
,V
. For a general
SV model, we obtain the following system discretizing (
9.38
). Given
w
0
N
=
k
A
κ
,
F
m
where
B
:=
M
+
:=
k
f
+
and
f
i
=
0
, find
w
m
+
1
N
N
∈ R
such that for
m
=
0
,...,M
−
1,
B
w
m
+
1
N
F
m
,
≥
w
m
+
1
N
(9.40)
≥
0
,
)
B
w
m
+
1
N
F
m
=
(
w
m
+
1
N
−
0
,
+
k
A
SV
,
F
m
M
u
N
and
f
i
=−
a
SV
(w
0
,b
i
)
.
where
B
:=
M
:=
k
f
+
Example 9.6.2
We consider an American put with strike
K
=
100 and maturity
T
=
0
.
5 within the Heston model, for which we set the parameters
α
=
2
.
5,
β
=
0
.
5,
ρ
=−
0. Figure
9.3
depicts the option prices as well as the
behavior of the free boundary at
t
0
.
5,
m
=
0
.
06,
r
=
=
T
.
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