I have been told that the Euler discretisation is exact for the GBM process.Is it true and how can I proof this? This would mean, for a GBM process, if I am increasing my discretisation step, the value is unaffected. However, in many application, the error for discretisation of GBM decreases with increasing steps. What is true?
It is exact if you shift to log coordinates first. In that case, you are discretising Brownian motion with drift i.e. $$ d\log S_t = (\mu-0.5\sigma^2 )dt + \sigma dW_t $$
It is definitely not exact in the original coordinates, since the probability of going negative is positive for this discretization.