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Appendix B
Parabolic Variational Inequalities
To formulate the parabolic variational inequalities (PVIs) for pricing European and
American contracts, we require suitable function spaces.
Let
V
,
H
be Hilbert spaces with norms
·
V
and
·
H
, respectively. As in
Appendix
A
, we assume that
V
→
H
with dense injection and identity
H
with its
H
∗
, so that we obtain the so-called evolution triple with dense injections,
V
dual
→
H
=
H
∗
→
V
∗
.
(B.1)
On the time interval
J
:=
(a, b)
⊂ R
, we introduce the spaces of “sum” and of
“intersection” type by
L
1
(J
L
2
(J
;
V
∗
),
S(a,b)
:=
;
H
)
+
(B.2)
;
H
∗
).
(B.3)
The present results on existence and time discretization of abstract parabolic evolu-
tion variational inequalities are based on [
5
].
L
2
(J
L
∞
(J
I(a,b)
:=
;
V
)
∩
B.1 Weak Formulation of PVI's
In the triple (
B.1
), we are given a linear, possibly non-selfadjoint operator
A
:
V
→
V
∗
with the associated bilinear form
a(u,v)
:=
A
u, v
V
∗
,
V
:
V
×
V
→ R
,
V
∗
×
V
where
·
,
·
V
∗
,
V
denotes the extension of the
H
innerproduct to
by continu-
ity. We assume that there are
α, β >
0
∀
∈
V
:|
|≤
V
V
,
u, v
a(u,v)
β
u
v
(B.4a)
2
2
V
∀
u
∈
V
:
a(u, u)
+
λ
u
H
≥
α
u
,
(B.4b)
for some
λ
≥
0.
A parabolic variational inequality (PVI) in strong form is given by a set
∅=
K
⊂
V
,
closed and convex with 0
∈
K
,
(B.5)
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