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is a martingale. Using a generalization of Proposition 1.2.7 for Lévy processes
(see, e.g. [40, Proposition 8.8]) it is sufficient to show that
E[
0
R
|
f(X
t
−
+
z)
−
2
ν(
d
z)
f(X
t
−
)
|
]
<
∞
.Wehave
2
2
ν(
d
z)
t
2
ν(
d
z) <
E
|
f(X
t
−
+
−
|
≤
|
|
|
|
∞
z)
f(X
t
−
)
T
sup
x
∂
x
f(x)
z
,
0
R
∈R
R
C
2
(
since
f
∈
R
)
has bounded derivatives and
ν
satisfies (
10.11
).
Remark 10.3.2
Using the Lévy-Khinchine representation (see Theorem
10.1.4
), we
derive from (
10.14
) the following formula for the action of the operator
A
on oscil-
lating exponents
e
iξ(x
+
z)
izξe
iξx
ν(
d
z)
1
2
σ
2
ξ
2
e
iξx
)e
iξx
γiξe
iξx
e
iξx
(
−
A
=
−
−
−
−
R
ψ(ξ)e
iξx
,
=
(10.15)
∈
S
(
R
)
, the spa
ce of fun
ctions in
C
∞
(
with
ξ
∈ R
.For
f
R
)
vanishing at infinity
faster than any negative power of
1
+|
x
|
2
, we can write
e
iξx
f(ξ)
d
ξ,
f(x)
=
(10.16)
R
(
2
π)
−
1
f(ξ)
e
−
iξx
f(x)
d
x
is the Fourier transform of
f
. By applying
(
10.15
) to the representation (
10.16
), using Fubini's theorem and the convergence
theorem of Lebesgue, we obtain
where
:=
R
e
iξx
) f(ξ)
d
ξ
f(ξ)
d
ξ.
ψ(ξ)e
iξx
(
−
A
f )(x)
=
(
−
A
=
(10.17)
R
R
Thus,
−
A
is a
pseudo-differential operator
(PDO) with symbol
ψ
.
Repeating the arguments which lead to Theorem 4.1.4 yields
C
1
,
2
(J
C
0
(J
Theorem 10.3.3
Let v
∈
× R
)
∩
× R
) with bounded derivatives in x
be a solution of
∂
t
v
+
A
v
−
rv
=
in J
× R
, T,x)
=
g(e
x
)
0
in
R
,
(10.18)
where
A
as in
(
10.14
)
with drift r
+
γ
.
Then
,
v(t,x) can also be represented as
e
−
r(T
−
t)
g(e
rT
+
X
T
)
v(t,x)
= E[
|
X
t
=
x
]
.
We again change to
time-to-maturity t
→
T
−
t
, to obtain a forward parabolic
e
rt
v(T
problem. Thus, by setting
u(t, s)
=:
−
t,x
−
(γ
+
r)t)
, we remove the drift
γ
and the interest rate
r
. Then,
u
satisfies
∂
t
u
J
u
−
A
=
0in
(
0
,T)
× R
,
(10.19)
g(e
x
)
in
with the initial condition
u(
0
,x)
=
R
and
1
2
σ
2
∂
xx
f(x)
J
f )(x)
(
A
=
+
(f (x
+
z)
−
f(x)
−
z∂
x
f (x))ν(
d
z).
(10.20)
R
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