Question: Let = 1 be the correlation process of two Brownian motions B1 and B2. Set B 1 = B1. Define B 2 by

Let ρ = ± 1 be the correlation process of two Brownian motions B1 and B2. Set Bˆ 1 = B1. Define Bˆ 2 by Bˆ 20 = 0 and dBˆ 2 = 1

√1 −ρ2 (dB2 −ρ dB1).

Show that Bˆ 1 and Bˆ 2 are independent Brownian motions. Note: Obviously this reverses the process of the previous exercise. It gives us dB2 = ρ dBˆ 1 +

1− ρ2 dBˆ 2 , so ρ dB1 can be viewed as the orthogonal projection of dB2 on dB1 = dBˆ 1.

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