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In the proof of Proposition 8.1.2, we already showed that
M
t
is a martingale. We
=
0
f(Z
τ
−
))J(
d
τ,
d
ζ)
is a martin-
gale. Indeed (compare also with the proof of Proposition 10.3.1),
show that
M
t
<c
(f (Z
τ
−
+
ς
s
(Z
τ
−
,ζ))
−
|
ζ
|
t
2
ν(
d
ζ)
d
τ
f(Z
τ
−
+
f(Z
τ
−
)
E
ς
s
(Z
τ
−
,ζ))
−
0
|
ζ
|
<c
t
2
ν(
d
ζ)
d
τ
2
≤
C
max
1
sup
z
∈R
d
|
∂
z
i
f(z)
|
E
|
ζ
|
<c
|
ς
s
(Z
τ
−
,ζ)
|
≤
i
≤
d
0
≤
C
t
2
)
d
τ
<
2
min
{
1
,
|
ζ
|
}
ν(
d
ζ)
E
(
1
+|
Z
τ
−
|
∞
,
|
ζ
|
<c
0
where we used (
15.2
) and (
15.3
) and the fact that
ζ
2
ν(
d
ζ) <
1
∧
∞
.The
R
same considerations can be made to show that
M
t
is a martingale, where we em-
ploy (10.11).
We repeat the arguments which lead to Theorem 4.1.4 and obtain, using Propo-
sition
15.2.1
, a generalization of Theorem 8.1.3
d
Theorem 15.2.2
Let d
:=
n
v
+
1,
an
d l
et G
⊆ R
be the state space of the pro-
cess Z
.
Let v
∈
C
1
,
2
(J
d
)
∩
C
0
(J
d
) with bounded derivatives in z
=
× R
× R
(x, y
1
,...,y
n
v
) be a solution of
∂
t
v
−
A
v
+
rv
=
v(
0
,z)
=
g(e
x
)
0
in J
×
G,
in G,
(15.12)
with
A
as in
(
15.2.1
).
Then
,
v(t,z)
=
V(T
−
t,z) can also be represented as
= E
e
−
r(T
−
t)
g(e
X
T
)
z
.
V(t,z)
|
Z
t
=
Consider the Bates models. It follows by (
15.4
)-(
15.7
) that the generator
A
=:
A
B
is given by
n
v
n
v
n
v
1
2
1
2
B
f )(z)
:=
β
i
y
i
∂
y
i
y
i
f(z)
(
A
y
i
∂
xx
f(z)
+
β
i
ρ
i
y
i
∂
xy
i
f(z)
+
i
=
1
i
=
1
i
=
1
r
1
2
+
λ
i
κ
y
i
∂
x
f(z)
n
v
n
v
+
−
λ
0
κ −
+
α
i
(m
i
−
y
i
)∂
y
i
f(z)
i
=
1
i
=
1
λ
0
+
λ
i
y
i
n
v
f(x
f(z)
ν
0
(
d
ζ),
+
+
ζ,y
1
,...,y
n
v
)
−
R
i
=
1
(15.13)
B
H
with
ν
0
as in (10.7). If
n
v
=
1 and
λ
0
=
λ
1
=
0, then
A
reduces to
A
in (9.12).
S
As a second example, we give the generator
A
:=
A
of the BNS model. From
(
15.10
) we deduce
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