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The Milshtein scheme [44] is obtained by adding the term
from the Itô-Taylor expansion () to the Euler scheme ().
|
(7.6) |
The same scheme is found for the Stratonovich interpretation from the Stratonovich-Taylor expansion () with ()
|
(7.7) |
where
The addition of this term increases the order of strong convergence from for the Euler scheme to 1 for the Milshtein scheme. It corresponds to that of the deterministic Euler scheme without any noise, that is with .
Thus, the Milshtein scheme can be interpreted as the proper generalization of the deterministic Euler scheme for the strong order convergence criterion ().
The generalization for our multidimensional Langevin equation () gives
|
(7.8) |
For the noise induced drift term we get
The additional stochastic term of the Milshtein scheme () corresponds to the drift term of the Euler scheme in Stratonovich interpretation () [45].
In the Euler scheme this term is
whereas in the Milshtein scheme it reads as
So, the term
is replaced by its mean value
, which is according to (). This small modification accounts for the difference in the order of convergence. However, if one is interested only in computing the moments
, for example, it can be shown, that the Euler and Milshtein algorithm are of equal accuracy. However, since both algorithms have approximately the same computational complexity, it does not seem to be justified to use the poorer Euler algorithm instead of the Milshtein algorithm.
Next: 7.2.3 Heun scheme
Up: 7.2 Stochastic integration schemes
Previous: 7.2.1 Euler scheme
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Werner Scholz
2000-05-16