Glasserman's "Monte Carlo Methods in Financial Engineering" on p. 265 states that the geometric Brownian motion can be modelled with :
$$S(t_n)=S(0) \exp(\sum_{i=1}^n X_i)$$
where $X_i$ are iid. I haven't been able to find an explanation for that, and am familiar with a completely different method of simulating a Brownian motion $S$. Where does this come from?
This is Ito's lemma. If $$ dS_{t}=a_{t}S_{t}dt+b_{t}S_{t}dW_{t}, $$ then $$ d(\log S_{t})=\frac{1}{S_{t}}dS_{t}-\frac{1}{2}\frac{1}{S^{2}}d[S,S]_{t}=\left(a_{t}-\frac{1}{2}b_{t}^{2}\right)dt+bdW_{t}. $$ Therefore, $$ \log S_{T}-\log S_{0}=\int_{0}^{T}a_{t}-\frac{1}{2}b_{t}^{2}dt+\int_{0}^{T}b_{t}dW_{t}. $$ Or, moving terms around and exponentiating, $$ S_{T}=S_{0}\exp\left(\int_{0}^{T}a_{t}-\frac{1}{2}b_{t}^{2}dt+\int_{0}^{T}b_{t}dW_{t}\right). $$
Of course, the lower and upper bounds of integration in the above do not have to be $0$ and $T$; they can be $t_{n-1}$ and $t_n$ for any $t_{n-1}<t_n$. Assuming now that $a_{t}=a$ and $b_{t}=b$ are constant, we can split time up into equally spaced intervals $t_{0}<t_{1}<\cdots<t_{n}$ with $$ S_{t_{i}}=S_{t_{i-1}}\exp\Bigg(\underbrace{\int_{t_{i-1}}^{t_{i}}a-\frac{1}{2}b^{2}dt+\int_{t_{i-1}}^{t_{i}}bdW_{t}}_{X_i}\Bigg) $$ and use the additive property of $\exp$ to arrive at the desired result.