These equations describe the correlation in time of geometric Brownian motion. When looking at a two moments in time (t and T) of a specific price-path we can observe two basic facts when comparing the paths between now and t and T respectively.

• the part up to t of the two paths are identical
• the part after t is uncorrelated with the part up to t

Using these two fact, the results are easy to verify.

List of symbols

	Value of the geometric Brownian motion at time t
	Value of the geometric Brownian motion at time T (with T>t)
	Variance
	Standard deviation
	Covariance
	Natural logarithm

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