From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: Noise (random) terms in ODEs
Date: 2 Feb 2001 15:29:20 GMT
Newsgroups: sci.math.num-analysis
Summary: Numerical solutions of stochastic differential equations

In article <m2elxifuem.fsf@pcmaxwell.intec.rug.ac.be>,
 Alberto BARSELLA <ishark@lsh01.univ-lille1.fr> writes:
|> Hi,
|>         I'm integrating a simple set of ODEs, and I need to introduce
|> some random terms into the rhs.  Up to now, I've been using a
|> variable-step algorithm (8th order Dormand-Prince), but I guess that
|> the stepsize control will not like at all a random variation of the
|> rhs..... I suppose that the problem of random terms in ODEs has
|> already been solved, any pointers to what I should read to learn about
|> what to do?

you are right, the stepsize mechanism would indeed not like noise and react
by reducing the stepsize to become horribly small.
what you are encountering now is known as a stochastic differential equation
there is much research in this field and some literature:
a search in Zentralblatt fuer Mathematik 
with stochastic; differential, equations
gives 275 hits
and here are some:

924.65146
Burrage, K.; Burrage, P.M.
General order conditions for stochastic Runge-Kutta methods for both 
commuting and non-commuting stochastic ordinary differential equation systems.
(English)
[J] Appl. Numer. Math. 28, No.2-4, 161-177 (1998). [ISSN 0168-9274] 
http://www.elsevier.nl/locate/apnum/

Lazakovich, N.V.
Finite difference approximations of a stochastic differential equation in 
Ito form. (Russian. English summary)
[J] Izv. Akad. Nauk Belarusi, Ser. Fiz.-Mat. Nauk 1996, No.2, 22-29 (1996). 
[ISSN 0002-3574]

Castell, Fabienne; Gaines, Jessica
The ordinary differential equation approach to asymptotically 
efficient schemes for solution of stochastic differential equations. (English)
[J] Ann. Inst. Henri Poincare, Probab. Stat. 32, No.2, 231-250 (1996). 
[ISSN 0246-0203]

930.93071
Kloeden, Peter E.; Keller, Hannes; Schmalfuss, Bjoern
Towards a theory of random numerical dynamics. (English)
[CA] Crauel, Hans (ed.) et al., Stochastic dynamics. Conference on Random 
dynamical systems, Bremen, Germany, April 28 - May 2, 1997. 
Dedicated to Ludwig Arnold on
the occasion of his 60th birthday. New York, NY: Springer. 259-282 (1999). 
[ISBN 0-387-98512-3]


Kloeden, Peter E.; Schmalfuss, Bjoern
Nonautonomous systems, cocycle attractors and variable time-step 
discretization. (English)
[J] Numer. Algorithms 14, No.1-3, 141-152 (1997). [ISSN 1017-1398]


Yannios, N.; Kloeden, P.E.
Time discretization solution of stochastic differential equations. (English)
[CA] May, Robert L. (ed.) et al., Computational techniques and 
applications: CTAC 95. Proceedings of the 7th biennial conference, 
Swinburne Univ. of Technology,
Melbourne, Australia, July 3--5 1995. Singapore: World Scientific. 823-830 
(1996). [ISBN 981-02-2820-1]

Kloeden, P.E.; Platen, E.; Hofmann, N.
Extrapolation methods for the weak approximation of Ito diffusions. (English)
[J] SIAM J. Numer. Anal. 32, No.5, 1519-1534 (1995). [ISSN 0036-1429]


Kloeden, P.E.; Platen, E.
Numerical methods for stochastic differential equations. (English)
[CA] Kliemann, Wolfgang (ed.) et al., Nonlinear dynamics and stochastic 
mechanics. Dedicated to Prof. S. T. Ariaratnam on the occasion of 
his sixtieth birthday. Boca Raton,
FL: CRC Press. CRC Mathematical Modelling Series. 437-461 (1995). 
[ISBN 0-8493-8333-1/hbk]


hope this helps
peter