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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?

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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.

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