Question: Itō's lemma and lognormal mean

I have always wanted to know why the lognormal distribution in the Black & Scholes (BS) option pricing model

is assumed to have a mean of  T*(r-0.5*sigma^2 ) . I have seen various explanation but non that make any sense.

I know that the mean in a lognormal distribution is exp(u+0.5*sigma^2) but why does it suddenly become

T*(r-0.5*sigma^2 ) in the BS model ?   I can understand the T = mean proportional to the time increament but why does

we switch sign from + to - .  I think it has to do with something called Ito's lemma but I am not sure ?


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