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14.4 Pricing Equation
As before we assume the risk-neutral dynamics of the underlying asset price is given
by
S
0
e
rt
+
X
t
,i
S
t
=
1
,...,d,
where
X
is a
d
-dimensional Lévy process with characteristic triplet
(
Q
,ν,γ)
under
a non-unique EMM. As shown in Lemma
14.1.3
, the martingale condition implies
γ
j
=−
Q
jj
2
=
e
z
j
z
j
ν
j
(
d
z),
−
−
1
−
j
=
1
,...,d.
R
We again show that the value of the option in log-price
v(t,x)
is a solution of a
multidimensional PIDE.
d
Proposition 14.4.1
Let X be a Lévy process with state space
R
and characteristic
triplet (
Q
,ν,γ) where the Lévy measure satisfies
(
14.15
).
Denote by
A
the integro-
differential operator
1
2
tr
D
2
f(x)
γ
∇
(
A
f )(x)
=
[
Q
]+
f(x)
d
f(x
z
∇
x
f(x)
ν(
d
z),
+
+
z)
−
f(x)
−
(14.18)
R
C
2
(
d
) with bounded derivatives
.
Then
,
the process M
t
for functions f
∈
R
:=
−
0
(
f(X
t
)
A
f )(X
s
)
d
s is a martingale with respect to the filtration of X
.
Proof
Let
Σ
=
(
Σ
ij
)
1
≤
i,j
≤
d
be given such that
ΣΣ
=
Q
. Proceeding as in the
proof of Proposition 10.3.1, we obtain using the Itô formula for multidimensional
Lévy processes and the Lévy-Itô decomposition
d
d
1
2
∂
x
i
f(X
t
−
)
d
X
t
+
d
f(X
t
)
=
1
Q
ij
∂
x
i
x
j
f(X
t
)
d
t
i
=
1
i,j
=
d
X
t
∂
x
i
f(X
t
−
+
f(X
t
)
−
f(X
t
−
)
−
)
i
=
1
d
d
Σ
ij
d
W
t
γ
∇
=
+
f(X
t
−
)
d
t
∂
x
i
f(X
t
−
)
i
=
1
j
=
1
∂
x
i
f(X
t
−
)
d
z
i
J
X
(
d
t,
d
z)
+
R
d
\{
0
}
i
=
1
f(X
t
−
+
z)
−
f(X
t
−
)
1
2
tr
[
Q
D
2
f(X
t
)
]
+
d
t
+
R
d
\{
0
}
z
i
∂
x
i
f(X
t
−
)
J
X
(
d
t,
d
z)
d
−
i
=
1
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