Let B1 and B2 be independent Brownian motions and dZ def

=



dZ1 dZ2



=



σ11 σ12

σ21 σ22dB1 dB2

 def

= AdB for stochastic processes σij, where A is the matrix of the σij.

(a) Calculate

a, b, and c with a > 0 and c > 0 such that LL = AA

, where L =



a 0 b c

.

(b) Define Bˆ = (Bˆ 1 Bˆ 2) by Bˆi0 = 0 and dBˆ = L−1AdB. Show that Bˆ 1 and Bˆ 2 are independent Brownian motions.

(c) Show that dZ = LdBˆ.

Note: This illustrates the implementation of Gram-Schmidt orthogonalization via the Cholesky decomposition discussed in Sections 3.9 and