Brownian motion (1 Viewer)

He-Mann

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If B(s) and B(t) were independent, then their sum would be B(s+t). However, when dependence exists, why is there an increase of 2s to the variance? Why does the variance increase in this manner?

Can someone give an intuitive explanation (possibly relating it to concrete things like stock prices?) of why the variance is 3s+t?
 
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seanieg89

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Handwavy explanation using a particle moving with Brownian motion:

Note B(t) is the particles position at some intermediate time t. If it ends up at B(s), then B(t) has a tendency to be in the same direction as B(s) from the origin (this is the dependence statement). This means B(s)+B(t) will typically have higher magnitude than B(s+t), where we simply randomly wander for s+t time. The variance statement quantifies this.
 
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He-Mann

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Handwavy explanation using a particle moving with Brownian motion:

Note B(t) is the particles position at some intermediate time t. If it ends up at B(s), then B(t) has a tendency to be in the same direction as B(s) from the origin (this is the dependence statement). This means B(s)+B(t) will typically have higher magnitude than B(s+t), where we simply randomly wander for s+t time. The variance statement quantifies this.
Thanks for giving me a new perspective on Brownian motion. :)
 

seanieg89

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No worries, hope it helped :). I haven't actually worked with Brownian motion before but some aspects of it seem like they would be pretty intuitive.
 

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