This is my first post so I am open for an suggestions in formating improvement. For some reason I can not find suitable literature for the following problem
What I want to do is calculate the option value of a basket defined through $d$ different assets of the stock market defined through geometric brownian motions $S = S_1(t), \dots, S_d(t)$, that satisfy the following SDE:
$ dS_i(t) = \mu_i S_i(t) dt + \sum_{j=1}^{d} \sigma_{ij} S_i(t) dW_j(t), \quad t \in [0,T], i = 1,\dots, d $
where $\mu_i$ are drift parameters, $(\sigma_{ij})_{ij}\in \mathbb{R}^{d\times d}$ is the defining a volatility matrix and $W_i(t)$ are independent brownian motions on a probability space. Furthermore we have that $E(dW_i) = 0$ and $E(dW_i^2) = dt$ and furthermore $E(dW_i dW_j)= \rho_{ij}dt$.
In the literature I found that "its easy" to show that through integration the system can be transformed for $i=2$ to: $ S_1(t) = S_1(0) \exp\left[ \left( \mu_i - \dfrac{\sigma_{11}^{2}+\sigma_{12}^{2} }{2} \right) t + \sigma_{11}W_1(t) + \sigma_{12} W_2(t)\right]$
$S_2(t) = S_2(0) \exp \left[\left(\mu_i - \dfrac{\sigma_{21}^{2}+\sigma_{22}^{2} }{2} \right) + \sigma_{21}W_1(t) + \sigma_{22} W_2(t) \right]$
Question: What are the values of the system of integrated geometric brownian motions in the general case, (or e.g. $i = 10$).
I found in a script of the KIT that its possible to model a European basket through the geometric brownian Motion
$dS(t) = \mu S dt + diag(\sigma) diag(S) L dW(t)$ with $S(t_*) = S_*$ where $\sigma = (\sigma_1, \dots, \sigma_d)$, $\mu > 0$, $ S = (S_1, \dots, S_d)$ , $dW = (dW_1, \dots, dW_d)$ and $L$ is the lower trianagle matrix which can be derived through the colesky decomposition of $\Sigma = L L^T$, while
$\Sigma := \left(\begin{matrix}1&\rho_{12}&\cdots&\rho_{1d}\\\rho_{21}&1&\ddots&\vdots\\\vdots&\ddots&1&\rho_{d-1d}\\\rho_{d1}&\cdots&\rho_{dd-1}&1\end{matrix}\right)$.
Question: Can someone help me to derive that system of geometric BM please?