TD - Itos lemma, Black-Scholes equation and a perpetual option

Question
In this question YOU MAY ASSUME
(i) that small changes df in the function f(S,t) are related to small changes
in S and t by Taylor’s theorem so that
df = fSdS + ftdt +
1
2
fSSdS2 + fStdSdt +
1
2
fttdt2 + · · ·
(ii) that S follows the lognormal random walk
dS
S
= rdt + σdX
where r and σ are constants and X is a random variable,
(iii) that dX2 → dt as dt → 0.
(a) Derive Itˆo’s lemma in the form
df = σSfSdX +
µ
ft + rSfS +
1
2
σ
2S
2
fSS
¶
dt
and comment briefly on whether or not your derivation is rigorous.
(b) Denote the fair value of an option by V (S,t). By constructing a portfolio
Π = V − ∆S where ∆ is to be determined, show that V satisfies the
Black-Scholes equation
Vt +
1
2
σ
2S
2VSS + rSVS − rV = 0.
(c) A PERPETUAL option is one whose value does not depends upon time.
Find the most general solution for the value of a perpetual option and
show that the value of a perpetual Put is given by
V = AS−2r/σ2
where A is a constant that depends on the specific details of the option.