GBM formulas - when to subtract 0.5σ^2 from u

Background – I often get confused when (not) to subtract. Here’s a brief summary.

The standard GBM dynamic is

                dS = mu S dt + σ S dW .... where mu and σ are time-invariant.

The standard solution is to find the dynamics of logS, denoted L,

                dL = (mu – 0.5σ² ) dt + σ dW ...  BM, not GBM. No L on the RHS.

               

                L (time=T)     ~  N (mean = (mu – 0.5σ² )T, std = .... )

So it seems our mu can’t get rid of –0.5σ² thingy ... Until we take expectation of S(time=T)

                E S(time=T) = S(time=0) exp(mu*T)     ... no σ² term

When we write down the Black Scholes PDE we use mu without the –0.5σ² thingy.

BS formula uses mu without the –0.5σ² thingy.