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Re: About stochastic differential equations
*To*: mathgroup at smc.vnet.net
*Subject*: [mg19578] Re: About stochastic differential equations
*From*: Didier Pieroux <dpieroux at ulb.ac.be>
*Date*: Sat, 4 Sep 1999 01:34:20 -0400
*Organization*: Free University of Brussels (ULB)
*References*: <7qkqvl$h4t@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Dear Dimitris,
> Blimbaum Jerry pointed an easier way to solve the problem (which
> I did not test because I had already implemented the notebook
> of David):
> soln = NDSolve[{x'[t] == Random[] + Sin[t], x[0] == 1}, x[t], {t, 0,
> 5}] // First
> Plot[x[t] /. soln, {t, 0, 5}, PlotRange -> All];
While this proposition seems correct at first sight, it is not. The
problem with stochastic differential equations (SDEs) is that the random
term is not analytical, and its derivative doesn't exist. In other
words, SDEs cant be treated like ODEs. If you want to use the SDE
formalism, you have to replace the random term by a Wiener process and
write your equation like
dx[t] == Sin[t] dt + dW(t)
where dW(t) accounts for the Wiener process. The problem is that dW(t)
is proportional to Sqrt[dt]. (see [1]).
So the equation modeling what you really want is
dx[t] == Sin[t] dt + A Random[] Sqrt[dt]
with A being the strength of the random term.
Therefore, the solution proposed by Jerry doesn't work because NDSolve
uses a variable step size method. About the version of David, I think
it suffers from the same problem because it also uses NDSolve the same
way (... I am perhaps missing something here because I still use
Mathematica 3 and I had to read its notebook from the text version :-( )
As conclusion, by integrating a SDE like an ODE with a random term
included, you will get a wrong result:
- if you use a constant step size method, the effective noise amplitude
will depends on the step size;
- if you use a variable step size method, the effective noise amplitude
will vary along the integration.
A direct consequence of this is that by increasing the precision of the
integration, you also change the amount of noise in your system
(Corollary: its statistical behavior will probably not converge as the
integration precision is increased...).
To know more about SDE, I often see [2] recommended. About numerical
integration, I have no idea (but have a look at
http://www.amazon.com/exec/obidos/Subject=Stochastic%20differential%20equati)
Regards
Didier
[1] Gardiner, "Handbook of Stochastic Methods", Springer-Verlag, ISBN
3-540-15607-0
[2] B.K. Oksendal, B.K. Ksendal, "Stochastic Differential Equations : An
Introduction With Applications (Universitext)", Springer Verlag ISBN:
3-540-63720-6
--
_________________________________________________________________
Didier Pieroux
Theoretical Nonlinear Optics, CP 231
Physics Department, Universite Libre de Bruxelles
Bvd du Triomphe, B-1050 Brussels, Belgium
Phone: ++ 32 2 650 5903, Fax: ++ 32 2 650 5824
http://www.ulb.ac.be/polytech/soa/IAP/iap.html
_________________________________________________________________
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