Since the Wiener process fluctuates around its expectation 0 it
can be approximated by means of symmetric random walks. As for
random walks we are interested in stochastic processes in
continuous time which are growing on average, i.e. which have a
trend or drift. Proceeding
from a Wiener process with arbitrary
(see Section
5.1) we obtain the generalized Wiener process
with drift rate
and variance
 |
(6.3) |
The general Wiener process
is at time
N
-distributed. For its increment in a small time
interval
we obtain
For
use the
differential notation:
 |
(6.4) |
This is only a different expression for the relationship
(5.3) which we can also write in integral form:
 |
(6.5) |
Note, that from the definition of the stochastic integral it
follows directly that
The differential notation (5.4) proceeds from the
assumption that both the local drift rate given by
and the
local variance given by
are constant. A considerably
larger class of stochastic processes which is more suited to model
numerous economic and natural processes is obtained if
and
in (5.4) are allowed to be time and state
dependent. Such processes
which we
call Itô-processes, are defined as solutions of stochastic
differential equations:
 |
(6.6) |
Intuitively, this means:
i.e. the process' increment in a small
interval of length
after time
is
plus a random fluctuation which is
N
distributed. A precise
definition of a solution of (5.6) is a stochastic
process fulfilling the integral equation
 |
(6.7) |
In this sense (5.6) is only an abbreviation of
(5.7). For
it follows immediately:
Since the increment of the Wiener process between
and
does not dependent on the events which occurred up to time
,
it follows that an Itô-process is Markovian.
Discrete approximations of (5.6) and (5.7)
which can be used to simulate Itô-processes are obtained by
observing the process between 0 and
only at evenly spaced
points in time
With
and
we get
or rather with the abbreviations
with identical independently distributed
N
-random variables
SFESimCIR.xpl
,
SFMSimOU.xpl