Use of Ito's lemma in ItosLemma.m (or any other method in Mathematica)

Another possibility with built-in functions:

(* Define the process *)
proc = ItoProcess[ⅆs[t] == μ s[t] ⅆt +
 σ s[t] ⅆw[t], s[t], {s, s0}, {t, 0}, 
 w \[Distributed] WienerProcess[]]

Now the process can be used as:

Expectation[z, z \[Distributed] proc[t]]
(* E^(t μ) s0 *)

Simplify[Expectation[Log[z], z \[Distributed] proc[t]], 
  Assumptions -> {σ > 0}]
(* t (μ - σ^2/2) + Log[s0] *)

A comment from @Diogo brought me on the right track:

<<ItosLemma`
dS=ItoMake[S[t],μ *S,σ*S] 
(* dt S μ+S σ Subscript[dB, 1] *)

ItoD[Log[S[t]]]
(* dt (μ-σ^2/2)+σ Subscript[dB, 1] *)