I see a bit of confusion.
Let's say you have a model of $n$ stochastic processes $X_i$. Let us assume that they follow a GBM. This means their dynamics are:
$$dX_t^i = \mu^i_t X_t^i dt + \sigma^i_t X_t^i dW_t^i,$$
where $W_t^i, \ i \in \{1, ..., n\}$ are $n$ correlated standard Brownian motion.
In this setting let us assume (without loss of generality) that $\mu$ and $\sigma$ do not depend on time. We thus have a vector
$$\mu = \{\mu^1, \mu^2, ..., \mu^n\}$$
and the variance covariance matrix $\Sigma$ (please spare me to write it in TeX).
Now, let us move to the simulation. Since (I believe) you have to simulate this in a computer, the setting has to be discretized.
Let us fix a time horizon $[0, T]$ and let us assume to have a tenor structure
$$\tau = \{t_0 = 0, \ t_1, \ t_2, ..., t_k = T \}.$$
The trick here is to observe that we can generate each process with a (composed) Euler scheme:
$$X_{t_{j+1}}^i = X_{t_{j}}^i \cdot e^{(\mu^i - \frac{1}{2}\sigma^2_i)(t_{j+1} - t_{j}) + \sigma_i \Delta W^i(t_{j+1})}.$$
To clarify notation
$$ \Delta W^i(t_{j+1}) = W^i_{t_{j+1}} - W^i_{t_{j}} \ \sim N(0, \ t_{j+1} - t_j)$$
is our increment for the $i$-th brownian.
As you can see, the correlation between assets is encapsulated in the brownian increments. Therefore, the problem of generating correlated assets (processes) boils down to a problem of generating correlated normal random variables.
Here it is where Cholesky decomposition is used. Indeed, call $L$ the Cholesky decomposition of $\Sigma$. It will be a $n$ square matrix.
The steps to construct correlated (with $\Sigma$) brownian increments are:
- For each time step, simulate $n$ random variables $Z^I$.
- Multiply them by $\sqrt{t_{j+1} - t_j}$ to obtain the wanted variance
- You get
$$\Delta W^i(t_{j+1}) = \sum_{q = 1}^nL_{i,q}Z_q.$$
As you can see, the correlation is on the stochastic driver and NOT on the drift. The drift is a deterministic movement.
Hope this clarifies.
