I have been struggling with the following problem: If you want to find a numerical result by simulating the paths of a stochastic differential equation, in particular a geometric brownian motion I have found literature saying you should go for the exact solution (continuous time) and literature stating that you should approximate the GBM (e.g. with the Euler-Maruyama Scheme).
From what I understand, there are SDE's who can be modelled exactly, and SDE's who can't. What's true for a GBM?
Additional info:
From Glasserman, P. (2004). Monte Carlo Methods in Fin. Engineering:
"Chapter 3 gives examples of continuous-time stochastic processes which can be simulated exactly [...]" (p. 339). Chapter 3 then lists the GMB in 3.2 (pp. 93ff.).
From Günther & Jüngel (2010). Finanzderivate mit Matlab. :
given $$dS_t = rS_tdt + \sigma S_tdW_t$$
transform this with the Euler-Maruyama Method to
$$\Delta S_t = rS_t\Delta t + \sigma S_t \Delta W_t$$
where $$\Delta W_t = Z \cdot \sqrt{\Delta t}$$ and $$Z\sim\mathcal{N}(0,1)$$ (pp. 102f.)
I hope that helps clarifying my issue. If not I'll happily give more details. Thanks