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To model asset prices by stochastic processes, knowledge about past events up
to time
t
should be incorporated into the model. This is done by the concept of
filtration
.
X
X
t
F
={
F
:
≤
≤
}
Definition 1.2.2
(Natural filtration) We call
0
t
T
the
natural
of the filtration
F
={
F
X
X
t
filtration for X
if it is the completion with respect to
P
:
F
X
t
0
≤
t
≤
T
}
, where for each 0
≤
t
≤
T
,
=
σ(X
r
:
r
≤
s)
.
A stochastic process is called
càdlàg
(from French 'continue à droite avec des
limités à gauche') if it has càdlàg sample paths, and a mapping
f
:[
0
,T
]→R
is
said to be càdlàg if for all
t
it has a left limit at
t
and is right-continuous
at
t
. A stochastic process is called
predictable
if it is measurable with respect to
the
σ
-algebra
F
∈[
0
,T
]
, where
F
is the smallest
σ
-algebra generated by all adapted càdlàg
processes on
Ω
.
Asset prices are often modelled by
Markov processes
. In this class of stochas-
tic processes, the stochastic behaviour of
X
after time
t
depends on the past only
through the current state
X
t
.
[
0
,T
]×
Definition 1.2.3
(Markov property) A stochastic process
X
={
X
t
:
0
≤
t
≤
T
}
is
Markov
with respect to
F
if
E[
f(X
s
)
|
F
t
]=E[
f(X
s
)
|
X
t
]
,
for any bounded Borel function
f
and
s
≥
t
.
No arbitrage considerations require discounted log price processes to be martin-
gales, i.e. the best prediction of
X
s
based on the information at time
t
contained in
F
t
is the value
X
t
. In particular, the expected value of a martingale at any finite time
T
based on the information at time 0 equals the initial value
X
0
,
E[
X
T
|
F
0
]=
X
0
.
Definition 1.2.4
(Martingale) A stochastic process
X
={
X
t
:
0
≤
t
≤
T
}
is a
mar-
tingale
with respect to
(
P
,
F
)
if
(i)
X
is
F
adapted,
E[|
X
t
|]
∞
≥
(ii)
<
for all
t
0,
(iii)
E[
X
s
|
F
t
]=
X
t
P
-a.s. for
s
≥
t
≥
0.
There is a one-to-one correspondence between models that satisfy the
no free
lunch with vanishing risk
condition and the existence of a so-called
equivalent local
martingale measure
(ELMM). We refer to [54, 55] for details. The most widely
used price process is a
Brownian motion
or
Wiener process
. Its use in modelling log
returns in prices of risky assets goes back to Bachelier [4]. Recall that the
normal
distribution
(μ, σ
2
)
with mean
μ
and variance
σ
2
with
σ>
0 has the density
N
∈ R
1
√
2
πσ
2
e
−
(x
−
μ)
2
/(
2
σ
2
)
,
μ, σ
2
)
f
N
(x
;
=
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