Constructing Correlated Wiener Processes

Question

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

Answer 1

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 $\sum_{k=1}^n c_{ik}(t)c_{kj}(t) = (C_t^2)_{ij} \neq \rho_{ij}(t)$.