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3.3.2 Generating a stationary Gaussian Markov sequence
Solve the stochastic differential equation
with a stochastic ``driving'' process ,
assumed to be uncorrelated Gaussian noise, i.e. Gauss distributed about
, with
.
The general solution to this equation reads
Inserting and
one finds that
At any time , the values of belong to a stationary Gauss
distribution with
,
and the process
has the Markov property.
The integrals
are elements of a random sequence, with
Gauss distributed with zero mean and
for . Their variance is
Here is the resulting recipe
for generating a stationary, Gaussian Markov sequence:
EXERCISE:
Employ the above procedure to generate a Markov sequence
with a given . Check if the sequence
shows the expected autocorrelation.
Next: 3.3.3 Wiener-Lévy Process (Unbiased
Up: 3.3 Random Sequences
Previous: 3.3.1 Basics
Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001