In the chapter 3.7 you model the asset's real-world movement in terms of the equation:

log S_t = log S_0 + mu*t + sigma*sqrt(t)*N(0,1),

which, according to me, leads to

dS_t = S_t*[mu + (sigma^2)/2]dt + S_t*sigma*dWt.

On the other hand, in the chapter 5.6 you state that the movement is described by the geometric Brownian motion

dS_t = mu*S_t*dt + sigma*S_t*dWt.

How to reconcile the two? Do I understand correctly, that in the meantime you silently changed the definition of mu (which in 3.7 does not stand for the actual growth as there is also the "jagging condribition" 1/2 sigma^2).

Thanks in advance for clarification.