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Wing wrote:
> Hello everyone.
>
> I have a question about a standard Brownian motion, W(t). Is it
> possible to express dW? Because I would like to observe the small
> change of W over time and compare my simulation.
I don't know exactly what you mean by "express". In standard analysis,
dW is a "notation", rather than an actual infinitesimal random
variable. In non-standard treatments I have seen, dW really is an
infinitesimal random variable with mean zero and standard deviation
sqrt(dt). See, eg., Edward Nelson, "Radically Elementary Probability
Theory", Ann. of Math. Studies No. 117, Princeton U. Press, 1987. I
believe this is now freely downloadable from Nelson's website.
Presumably, your simulation uses a discrete-time approximation? Anyway,
discrete increments give W(t+h) - W(t) = N(0,h) (i.e., variance = h).
This is exact for all h > 0. (It assumes variance of 1.0 per unit time,
since you used the word "standard".) Successive increments are
independent, so W(t4) - W(t3) is independent of W(t2) - W(t1) for any
t1 < t2 <= t3 < t4.
R.G. Vickson
>
> Thank You.
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