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Chapter 10

C

Chirag Wadhwa

Member
In Continuous-time Brownian Motion Increments, what do white noise terms represent?
In the image below, how are the expressions "Cov(dZ_s,dZ_t)" and "Cov(dW_s,dW_t)" obtained?
Image Link:-https://ibb.co/gM9JKkX
 
The covariance of independent random variables is 0. The independent increments property of Brownian motion hence tell us the covariance of dW_t and dW_s is 0 if t<>s, because there isn't any overlap between the (infinitesimal) increments. If t=s then it's just the variance of dW_t, which is dt. I can only assume that somewhere Z is defined as sigma * W + some constant, which therefore explains why the variance of dW_t is sigma^2 * dt.
 
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