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d
,
+
A
−
=
× R
∂
t
u
u
ru
0in
(
0
,T)
(16.24)
d
.
=
R
u(T )
g
in
(16.25)
We set
t
0
=
and trans-
forming to time-to-maturity, we end up with the following parabolic evolution prob-
lem: find
u
0 for notational convenience. Testing with a function
v
∈
V
L
2
((
0
,T)
H
1
((
0
,T)
;
V
∗
)
s.t. for all
v
∈
;
V
)
∩
∈
V
and a.e.
t
∈[
0
,T
]
the following holds:
∂
t
u, v
V
∗
×
V
+
a(u,v)
=
0
,
(
0
)
=
g,
(16.26)
where
the bilinear form a(ϕ,φ)
d
)
is closely re-
lated to the Dirichlet form of the stochastic process X
. Although in option pric-
ing, only the homogeneous parabolic problem (
16.26
) arises, the inhomogeneous
equation (
16.27
) is useful in many applications. We mention only the compu-
tation of the time-value of an option, or the computation of quadratic hedging
strategies and the corresponding hedging error. Thus, we consider the nonhomo-
geneous analogue of the above equation. The general problem reads: Find
u
=−
A
ϕ,φ
V
∗
,
V
+
r(ϕ,φ)
L
2
(
R
∈
L
2
((
0
,T)
H
1
((
0
,T)
;
V
∗
)
s.t.
;
V
)
∩
∂
t
u, v
V
∗
×
V
+
a(u,v)
=
f, v
V
∗
×
V
in
(
0
,T),
∀
v
∈
V
u(
0
)
=
g
(16.27)
;
V
∗
)
. Now we consider the localization of the unbounded
problem to a bounded domain
G
. For any function
u
with support i
n
a bounded
domain
G
L
2
((
0
,T)
for some
f
∈
d
, we denote by
u
the zero extensions of
u
to
G
c
d
⊂ R
= R
\
G
and define
A
G
(u)
=
A
(
u)
. The variational formulation of the pricing equation on a bounded
d
L
2
((
0
,T)
H
1
((
0
,T)
V
G
)
∗
)
s.t. for all
domain
G
⊂ R
reads: Find
u
∈
;
V
G
)
∩
;
(
v
∈
V
G
and a.e.
t
∈[
0
,T
]
the following holds:
∂
t
u, v
V
G
×
V
G
+
a
G
(u, v)
=
f, v
V
G
×
V
G
,
(16.28)
u(
0
)
=
g
|
G
,
(16.29)
L
2
(G)
where
a
G
(u, v)
consist of func-
tions which vanish in a weak sense on
∂G
. A comparable result for general Feller
processes does not appear to be available, yet.
:=
a(
u,
v)
. The spaces
V
G
=: {
v
∈
:
v
∈
V
}
Remark 16.5.4
Formulation (
16.28
)-(
16.29
) naturally arises for payoffs with finite
support such as digital or (double) barrier options. The truncation to a bounded
domain can thus be interpreted economically as the approximation of a standard
derivative contract by a corresponding barrier option on the same market model.
Existence and uniqueness of weak solutions of (
16.28
)-(
16.29
) follows from
continuity of the bilinear form
a
G
(
·
,
·
)
and a Gårding inequality which follows from
the Theorem
16.5.5
.
Ψ
2
m
(x)
ρ,δ
Theorem 16.5.5
Let the generator
A
(x, D)
∈
be a pseudodifferential
operator of variable order
2
m
(x)
,0
<m
i
(x
i
)<
1,
i
=
1
,...,d with
m
(x)
=
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