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d
d
Σ
ij
d
W
t
=
(
A
f )(X
t
)
d
t
+
∂
x
i
f(X
t
−
)
i
=
1
j
=
1
f(X
t
−
))J
X
(
d
t,
d
z).
As already shown in Proposition 8.1.2,
i
=
1
∂
x
i
f(X
t
−
)
j
=
1
Σ
ij
d
W
t
+
(f (X
t
−
+
z)
−
d
R
is a martin-
C
2
(
d
)
and the Lévy measure
ν
satisfies (
14.15
), we have similar
gale. Since
f
∈
R
to Proposition 10.3.1 that also
f(X
t
−
))J
X
(
d
t,
d
z)
is a martin-
d
(f (X
t
−
+
z)
−
R
gale.
Repeating the arguments which lead to Theorem 4.1.4 yields
C
1
,
2
(J
d
)
C
0
(J
d
) with bounded derivatives in
Theorem 14.4.2
Let v
∈
× R
∩
× R
x be a solution of
∂
t
v
+
A
v
−
rv
=
d
, T,x)
=
g(e
x
1
,...,e
x
d
)
d
,
0
in J
× R
in
R
(14.19)
with
A
as in
(
14.18
)
with drift r
+
γ
.
Then
,
v(t,x) can also be represented as
e
−
r(T
−
t)
g(e
rT
+
X
T
)
|
X
t
=
x
.
v(t,x)
= E
We again change to
time-to-maturity t
→
T
−
t
and remove the drift
γ
and the
e
rt
v(T
interest rate
r
by setting
u(t, s)
=:
−
t,x
1
−
(γ
1
+
r)t,...,x
d
−
(γ
d
+
r)t)
.
14.5 Variational Formulation
For the variational formulation, we need anisotropic Sobolev spaces of fractional
order as defined in (13.4). Let
Q
=
(σ
i
σ
j
ρ
ij
)
1
≤
i,j
≤
d
, where
ρ
ij
is the correlation of
the Brownian motion
W
i
and
W
j
.Weset
ρ
=
(ρ
1
,...,ρ
d
)
with
1 if
σ
i
>
0
,
α
i
/
2 f
σ
i
=
ρ
i
=
0
,
with
α
giveninAssumption
14.3.4
. The variational formulation of the PIDE reads
L
2
(J
H
ρ
(
d
))
H
1
(J
L
2
(
d
))
such that
Find
u
∈
;
R
∩
;
R
a
J
(u, v)
H
ρ
(
d
),
a.e. in
J,
(∂
t
u, v)
+
=
0
,
∀
v
∈
R
(14.20)
u(
0
)
=
u
0
,
g(e
x
1
,...,e
x
d
)
and
form
a
J
(
·
,
·
)
H
ρ
(
R
d
)
where
u
0
(x)
:=
the
bilinear
:
×
H
ρ
(
d
)
R
→ R
is given by
1
2
a
J
(ϕ, φ)
ϕ)
Q
∇
:=
(
∇
φ
d
x
d
R
ϕ(x
z
i
ϕ
x
i
(x)
φ(x)ν(
d
z)
d
x.
d
−
+
−
−
z)
ϕ(x)
d
d
R
R
i
=
1
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