Let's assume we're happy with simulating $n$ stocks as geometric Brownian motion (GBM). But say we also want the prices to be correlated.
When I searched around for how to construct correlated paths, the typical strategy was to generate paths of Brownian motion with specified correlations, and then use those paths to construct GBM.
However, my calculations (and simulations) show that the correlation of the GBM is not the same as the correlation of the underlying Brownian motion, and the GBM correlations go to zero as time goes to infinity. This doesn't make much sense to me from a modeling perspective.
So in summary, is it even reasonable to use correlated Brownian motion to construct GBM, or would it make more sense in terms of modeling to somehow construct GBM paths with a constant and specified correlation coefficient?
Or did I just mess up with the analysis?
Please let me know if something isn't clear. Thanks.
EDIT: I understand how to create correlated paths of Brownian motion, specifically using the Cholesky decomposition. Here's what I know:
Say $\Sigma$ is a correlation matrix, and $LL^T$ is its Cholesky decomposition. Let $B(t)$ be $d$-dimensional Brownian motion. Then $L B(t)$ is also $d$-dimensional Brownian motion, and Corr$(B_i(t),B_j(t)) = \rho_{ij}$, where $\rho_{ij}$ is the $(i,j)$ element of the correlation matrix $\Sigma$.
Let $$X_i(t) = \mu_iX_i(t)dt + \sigma_i X_i(t)[LB(t)]_i$$ where $[LB(t)]_i$ is the $i^\text{th}$ element of $LB(t)$. Since we know that $[LB(t)]_i$ is Brownian motion, we know that $X_i(t)$ is geometric Brownian motion.
Here's the issue: Let's look at Corr$(X_i(t),X_j(t))$. We can use the result here to get Cov$(X_i(t),X_j(t))$. It says that $$\text{Cov}(X_i(t),X_j(t)) = X_i(0)X_j(0)e^{(\mu_1+\mu_2)t}\left(e^{\rho_{ij} \sigma_i\sigma_j}-1\right)$$
To get the correlation of the GBM paths, use the above fact and that $$\text{Var}(X_i(t))=X_i(0)^2 e^{2\mu_i t}\left(e^{\sigma_i^2 t}-1\right)$$
Thus we have $$\text{Corr}(X_i(t),X_j(t)) = \frac{e^{\rho_{ij}\sigma_i\sigma_j t}-1}{\sqrt{\left(e^{\sigma_i^2t}-1\right)\left(e^{\sigma_j^2t}-1\right)}}$$
Then (I am pretty sure this is always true) $$\lim_{t\to\infty}\text{Corr}(X_i(t),X_j(t)) = \begin{cases} 1, i=j\\ 0, i\neq j \end{cases} $$
To me at least, this isn't what we want. We would rather have the correlation be constant.
Let's instead try to get correlated GBM by using $$L X_t$$ where $X_t$ is $d$-dimensional GBM, and the 1D components are independent. There issue here is that I don't think $L X_t$ is GBM. In other words, the sum of independent GBM's is not GBM. We could try taking the product of correlated GBM's, but I haven't gone down that road yet. And before I do, I wanted to post this question.