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Construction

Hello. I'm reading the attached paper about the construction of correlated processes given a correlation matrix. But I am stuck on equation (2.23) -- surely it should say $c_{ik} . c_{kj} = \rho_{ij}$ instead?

I must be mistaken -- and would be grateful if someone could explain (2.23).

Thanks

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You're forgetting the transposition in the Cholesky decomposition. We have that $$\sum_{k=1}^n c_{ik}(t) c_{jk}(t) dt = \sum_{k=1}^n (C_t)_{ik} (C_t^*)_{kj} dt = (C_t C_t^*)_{ij} dt = \rho_{ij}(t) dt$$ The reason the indices have been reversed is that the second matrix in the matric product is transposed. Note that $\sum_{k=1}^n c_{ik}(t)c_{kj}(t) = (C_t^2)_{ij} \neq \rho_{ij}(t)$.

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  • $\begingroup$ Thank you Rhys that clears things up!!! $\endgroup$ – Henry P. Mar 23 at 14:04
  • $\begingroup$ This was very helpful! $\endgroup$ – Henry P. Mar 23 at 14:19

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