Definition 22.1
A
decomposition 
of the interval
![$ [a,b]$](sfehtmlimg4266.gif)
is
understood to be a set

of points

with

. Through this the interval
![$ [a,b]$](sfehtmlimg4266.gif)
is
decomposed into

sub-intervals
![$ [t_k,t_{k+1}],$](sfehtmlimg4269.gif)
where

,
that is, the length of the largest resulting sub-interval and is
referred to as the
refinement of the decomposition

.
Definition 22.2
For a function
![$ w :[a,b]\longrightarrow\mathbb{R}$](sfehtmlimg4273.gif)
and a
decomposition

one defines
the
variation of

with respect to

as:

is called the
total variation of

on
![$ [a,b]$](sfehtmlimg4266.gif)
. If

holds, then

is of
finite variation on
![$ [a,b]$](sfehtmlimg4266.gif)
.
Definition 22.3
Given the functions
![$ f,w : [a,b] \to \mathbb{R}$](sfehtmlimg4277.gif)
and a
decomposition

, choose for

partitions
![$ \tau_k\in[t_k,t_{k+1}]$](sfehtmlimg4279.gif)
and form:
If
$&tau#tau;$
converges for

to a
limiting value

, which does not depend on the chosen
decomposition

nor on the choice of the partitions

, then

is called the
Riemann-Stieltjes integral
of

. One writes:
For

we get the
Riemann Integral as a special case
of the Stieltjes Integrals.
Theorem 22.3 (Radon-Nikodym)
Let

and

be positive measures on

with
-
and
is absolutely continuous with respect to
, then from
it follows that
for all
(written:
).
When a non-negative

-measurable function

exists on

then it holds that:
In particular, for all measurable functions

it holds that:
Remark 22.1
One often uses the abbreviation

in the Radon-Nikodym
theorem and refers to

as the
density of

with respect to

.
Due to its construction

is also referred to as the
Radon-Nikodym derivative. In this case one writes

.
An important tool in stochastic analysis is the transformation of
measure, which is illustrated in the following example.
Example 22.1
Let

be independent random variables with standard
normal distributions on the measurable space

and

.
Then by
an equivalent probability measure

for

is defined. For
the distribution of the

under the new measure

it holds that:
in other words

are, with respect to

,
independent and normally distributed with expectations

and
![$ {\mathop{\text{\rm\sf E}}}_{{\rm Q}}[(Z_i-\mu_i)^2]=1.$](sfehtmlimg4332.gif)
Thus
the random variables

are independent random variables with standard normal
distributions on the measurable space

Going from
to
by multiplying by
changes the
expectations of the normally distributed random variables, but the
volatility structure remains notably unaffected.
The following Girsanov theorem generalizes this method for the
continuous case, that is, it constructs for a given
-Brownian
motion
an equivalent measure
and an appropriately
adjusted process
, so that it represents a
-Brownian
motion. In doing so the ("arbitrarily" given) expectation
is replaced by an ("arbitrarily" given) drift, that is, a
stochastic process
.
Theorem 22.4 (Girsanov)
Let

be a probability space,

a
Brownian motion with respect to

,

a filtration in

and

an adapted stochastic process. Then
defines a martingal with respect to

and

. The
process

defined by
is a Wiener process with respect to the filtration

and
 |
(22.1) |
is a

equivalent probability measure

.
The Girsanov theorem thus shows that for a
-Brownian motion
an equivalent probability measure
can be found such
that
, as a
-Brownian motion at time
, contains the
drift
. In doing so (A.1) means that:
for all
Remark 22.2
With the relationships mentioned above

is by all means a
martingale with respect to

and

when the
so-called
Novikov Condition
is met, that is, when

does not vary too much.
Another important tool used to derive the Black-Scholes formula by
means of martingale theory is the martingale representation
theory. It states that every
-martingale under certain
assumptions can be represented by a predetermined
-martingale by means of a square-integrable process.
Theorem 22.5 (Martingale Representation theorem)
Let

be a martingale with respect to the probability measure

and the filtration

, for which the volatility
process

of

almost surely

for
all
![$ t\in[0,T]$](sfehtmlimg4347.gif)
, where
![$ \sigma^2_t={\mathop{\text{\rm\sf E}}}_{{\rm Q}}[M^2_t\vert{\cal{F}}_t].$](sfehtmlimg4348.gif)
If

is another
martingale with respect to

and

, then there
exists (uniquely defined) on

an adapted stochastic
process

with

with:
Example 22.2
It is easy to show that the standard Wiener process

with
respect to the probability measure

is a martingale with
respect to

and its corresponding filtration

. If

is another martingale with respect to

and

,
then according to the previous theorem there exists a

adapted stochastic process

, so that
Remark 22.3
Writing the last expression in terms of derivatives:
The example shows once again that a martingale cannot possess a
drift.