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and it is symmetric around
μ
. Normality assumptions in models of log returns of
risky assets' prices imply the assumption that upward and downward moves of
prices occur symmetrically.
Definition 1.2.5
(Wiener process) A stochastic process
X
={
X
t
:
t
≥
0
}
is a
Wiener
process
on a probability space
(Ω,
F
,
P
)
if (i)
X
0
=
0
P
-a.s., (ii)
X
has indepen-
dent increments, i.e. for
s
≤
t
,
X
t
−
X
s
is independent of
F
s
=
σ(X
u
,u
≤
s)
,
(iii)
X
t
+
s
−
X
t
is normally distributed with mean 0 and variance
s>
0, i.e.
X
t
+
s
−
X
t
∼
N
(
0
,s)
, and (iv)
X
has
P
-a.s. continuous sample paths. We shall de-
note this process by
W
for N. Wiener.
In the Black-Scholes stock price model, the price process
S
of the risky asset is
modelled by assuming that the return due to price change in the time interval
t >
0
is
S
t
+
t
−
S
t
S
t
S
t
=
=
rt
+
σW
t
,
S
t
in the limit
t
→
0, i.e. that it consists of a deterministic part
rt
and a random part
σ(W
t
+
t
−
W
t
)
. In the limit
t
→
0, we obtain the stochastic differential equation
(SDE)
d
S
t
=
rS
t
d
t
+
σS
t
d
W
t
,
0
>
0
.
(1.1)
The above SDE admits the unique solution
S
t
=
S
0
e
(r
−
σ
2
/
2
)t
+
σW
t
.
This exponential of a Brownian motion is called the
geometric Brownian motion
.
The stochastic differential equation (
1.1
) for the geometric Brownian motion is a
special case of the more general SDE
d
X
t
=
+
σ(t,X
t
)
d
W
t
,X
0
=
b(t, X
t
)
d
t
Z,
(1.2)
for which we give an existence and uniqueness result.
Theorem 1.2.6
We consider a probability space (Ω,
F
,
P
) with filtration
F
and a
F
P
F
Brownian motion W on (Ω,
,
) adapted to
.
Assume there exists C>
0
such
that b, σ
: R
+
× R → R
in
(
1.2
)
satisfy
|
b(t, x)
−
b(t, y)
|+|
σ(t,x)
−
σ(t,y)
|≤
C
|
x
−
y
|
,x,y
∈ R
,t
∈ R
+
,
(1.3)
|
b(t, x)
|+|
σ(t,x)
|≤
C(
1
+|
x
|
),
x
∈ R
,t
∈ R
+
.
(1.4)
Assume further X
0
=
Z for a random variable which is
F
0
-measurable and satisfies
2
E[|
Z
|
]
<
∞
.
Then
,
for any T
≥
0, (
1.2
)
admits a
P
-a
.
s
.
unique solution in
[
0
,T
]
satisfying
E
sup
0
2
<
T
|
X
t
|
∞
.
(1.5)
≤
t
≤
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