Brownian motion

October 18, 2023

Brownian motion #

https://chat.openai.com/share/971a53a8-6c87-48d8-bed7-84d4975e2e60

\[ W_{N,t} = \sum_{i=1}^{Nt}{\frac{1}{\sqrt{N}}X_{i}} \]

We are considering N events in time t. For example, we are looking at 24 data points per day (at each hour) over multiple days.

Why are we dividing by \( \sqrt{N} \) but not \( \sqrt{Nt} \)?

In summary,

N: #

Represents the number of steps or discrete events. As \( N \) becomes large (approaching infinity for the continuous case), each step’s contribution is scaled down by a factor of \( \frac{1}{N} \) so that the cumulative effect remains stationary and well-behaved. In some real-world settings like stock markets, \( N \) might be finite but very large, reflecting the many small price changes that occur within a given time frame.

\( t \): #

• Reflects the time interval or length of observation. It’s not a model parameter in the same way \( N \) is. Instead, it’s an inherent dimension of the system being studied.
• The variance of the process grows linearly with \( t \), reflecting the cumulative effects over time. As you mentioned, we do not adjust or control for \( t \) in the same way we do for \( N \); we’re interested in its effects.
• The two—\( N \) and \( t \)—indeed belong to different conceptual categories. \( N \) is about discretization or granularity, whereas \( t \) is about duration or extent.

And you’re right in noting the similarity between \( N \) and \( t \) when looking at their cumulative effects: both give a linear growth in variance. But their roles and interpretations are distinct, with \( N \) being a model parameter (often about granularity or resolution) and \( t \) being an intrinsic dimension tied to the length of observation or duration.