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We refer to [131, Theorem 5.2.1] or [120, Theorem 2.3.1] for a proof of this
statement. Note that the Lipschitz continuity (
1.3
) implies the linear growth condi-
tion (
1.4
) for time-independent coefficients
σ(x)
and
b(x)
. For any
t
≥
0 one has
0
|
b(s, X
s
)
|
,
0
|
σ(s,X
s
)
|
2
d
s<
∞
-a.s., i.e. the solution process
X
is a
particular case of a so-called
Itô process
. Equation (
1.2
) is formally the differential
form of the equation
d
s<
∞
,
P
t
t
X
t
=
Z
+
b(s, X
s
)
d
s
+
σ(s,X
s
)
d
W
s
,
0
0
for
t
∈[
0
,T
]
. In the derivation of pricing equations, it will become important
to check under which conditions the integrals with respect to
W
,i.e.
0
φ
s
d
W
s
,
are martingales. The notion of stochastic integrals is discussed in detail in [120,
Sect. 1.5].
Proposition 1.2.7
Let the process φ be predictable and let φ satisfy
,
for T
≥
0,
2
d
t
<
∞
.
T
E
|
φ
t
|
(1.6)
0
,
M
t
:=
0
φ
s
d
W
s
is a martingale
.
Then
,
the process M
={
M
t
:
t
≥
0
}
For a proof of this statement, we refer to [131, Theorem 3.2.1]. In mathematical
finance, we are interested in the dynamics of
f(t,X
t
)
, e.g. where
f(t,X
t
)
denotes
the option price process. Here, the Itô formula plays an important role.
Theorem 1.2.8
(Itô formula)
Let X be given by the Itô process
(
1.2
),
and let
f(t,x)
C
2
(
∈
[
0
,
∞
)
× R
)
,
i
.
e
.
f is twice continuously differentiable on
[
0
,
∞
)
× R
.
Then
,
for Y
t
=
f(t,X
t
) we obtain
∂
2
f
∂x
2
(t, X
t
)
∂f
∂t
(t, X
t
)
d
t
∂f
∂x
(t, X
t
)
d
X
t
+
1
2
(
d
X
t
)
2
,
d
Y
t
=
+
·
(1.7)
where (
d
X
t
)
2
=
(
d
X
t
)
·
(
d
X
t
) is computed according to the rules
d
t
·
d
t
=
d
t
·
d
W
t
=
d
W
t
·
d
t
=
0
,
d
W
t
·
d
W
t
=
d
t.
We refer to [120, Theorem 1.6.2] for a proof of the Itô formula. A sketch of
the proof is given in [131, Theorem 4.1.2]. We note in passing that the smoothness
requirements on the function
f
in Theorem
1.2.8
can be substantially weakened.
We refer to [132, Sects. II.7 and II.8] and [40, Sect. 8.3] for general versions of the
Itô formula for Lévy processes and semimartingales.
1.3 Further Reading
An introduction to financial modelling and option pricing can be found in Wilmott
et al. [161] and the corresponding student version [162]. More details on risk-neutral
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