# Boost :

From: Manfred Doudar (manfred.doudar_at_[hidden])
Date: 2005-10-11 22:48:17

Deane Yang wrote:

>The application is to Brownian motion (which of course originated in
>physics!), which, to oversimplify, can be viewed as a time-dependent
>Gaussian distribution whose variance grows linearly with time and
>therefore its standard deviation grows proportionally to the square root
>of time. The proportionality constant in the standard deviation is
>called the volatility, and the units for volatility are the units for
>the standard deviation (think of that as a spatial dimension) divided by
>the square root of time.
>
>To take another tack, Brownian motion is closely related to the heat
>equation (again, a physical thing)
>
>u_t = (v^2/2)u_xx,
>
>where v is a constant measuring how well heat conducts in the material
>(I'll let the physicists make this more precise). The quantity v is the
>volatility.
>
>
>

Yes, and also used all over the place in geophysics and seismicity,
not to mention elsewhere, across the broad spectrum of physical sciences,
and as already mentioned by others, economics. ...You can go further,
and see applications in gridded FFT work, and data compression techniques.

... no one seemingly want's to mention the F-word, but we're talking
about Fractals, for purposes of statistical measure.

In short, the fractal dimension in physical sciences is commonly used
to measure the "distance" that "particles" need to travel/flow between
two points in (generally), some hyper-volume.

A readable paper on the subject - from an earth sciences point of view,
can be got from here:

>You could say that we should be working with v^2 and not v itself, and I
>won't disagree. But I have to live with common usage and convention.
>
>As for why Brownian motion matters in finance, I'm going to plead
>"off-topic" here, but you could try to look for explanations of the
>Black-Scholes formula. (Terence Tao at math.ucla.edu has a terrific but
>very terse explanation)
>
>So indeed, things like (time_duration)^(-1/2) do arise and are needed.
>Sorry. Your hopes were entirely reasonable, but fractional powers of
>fundamental units are really needed.
>
>

This touches on the volitility of timeseries in financial domains, and
fractional
powers, but for the less technically minded. -Again the concept of
distance of
travel applies here too.

>But couldn't this be done rather nicely using the MPL map type, using a
>compile-time rational number type? I'm not exactly sure how to contruct
>the derived unit type from the unit types of two factors, but I bet it
>could be done most easily using this approach.
>
>Matt Calabrese wrote:
>
>
>>On 10/11/05, Deane Yang <deane_yang_at_[hidden]> wrote:
>>
>>
>>

Cheers,

```--
Manfred Doudar
MetOcean Engineers
www.metoceanengineers.com
```