Sometimes I see the following in certain papers for SDE in $\mathbb{R}^n$:
$$dX = \mu dt + \sigma dB$$
But they specify $\mathbb{E}[B^i B^j] = D^{ij} \neq \delta^{ij}$ for some symmetric matrix $D$.
I have a suspicion that this is equivalent to the following SDE
$$dX = \mu dt + \sigma D dW$$
Where now $W_t$ is a Brownian motion that does in fact satisfy the usual uncorrelated properties. For example, this question seems to support the interchanging of correlations and just multiplying by a matrix.
Is this correct?
Consider $$B=AW$$ with $A$ a suitable matrix. If you calculate (using Einstein's summation convention) $$\mathbb{E}[B_iB_j] = A_{ik}A_{jl}\mathbb{E}[W_k W_l] = A_{ik}A_{jl}\delta_{kl}t = A_{ik}A_{jk}t$$ So it works if you take any matrix $A$ such that $D_{ij}=A_{ik}A_{jk}$.