伊藤の補助定理

金融工学の論文を読めば伊藤の補助定理がよく使われているから少し調べています。
It seems that Ito's lemma involves a function (F) of an underlying asset price (S(t)) and time (t). The asset price is stochastic (Brownian motion, martingale, etc). It seems that Ito's lemma relates the change in F (delta(F)) between times k-1 and k to the partial derivatives of F with respect to S and t. I am not sure why this is cited in expositions involving the Black-Scholes theorem, but it seems that F might be considered to be the price of an option or some other derivative.
The main value of Ito's lemma seems to be that it provides an expression for a differential dF which can be integrated to obtain F. Thus, if we want to compute an integral of some stochastic variable W, we can define a function F in terms of W and compute dF via Ito's lemma. Then integrating this term will, if we chose a suitable F, enable us to isolate the integral we wish to compute on the left side of the equation and be left without an integral on the right side. The value of Ito's lemma over the standard differential calculus is that it works for stochastic functions where dS/dt etc might not exist.
The basic expression for function F(S,t) is:
DF = (dF/ds)(delta S) + (dF/dt)(delta t) + (d^2F/dS^2)(delta S)^2
Where DF is really dF and dF/dS is the partial derivative of F with respect to S. The above can be simplified further, using
delta S = (a(dt) + sigma*(delta W))
In the text, "Black Scholes and Beyond", Ito's lemma is mentioned twice, and I do not comprehend its use in either case. On the first occasion, it is used to justify the mean return for an asset whose return follows Brownian motion with instantaneous return mu and variance sigma. The mean over time (T-t) is:
mean = (mu- (sigma^2/2))(T-t)
I assume that a mean for the stochastic process S is somehow found, but I am unsure how.
The second point is in the valuation of a portfolio for a stock which pays a dividend once during the lifetime of an option. The portfolio consists of the stock and a riskless zero coupon bond which matures on the ex-dividend date to D (dividend) and is sold short. The value of the portfolio (pi) is:
pi = S - e^(-r(t1-t))D
The volatility of this portfolio (sigma), via Ito's lemma,is derived as:
sigma = S/(S - e^(-r(t1-t))D)
I am uncertain of how Ito's lemma is used to derive volatility, although dF can be written in terms of sigma^2. I am not sure what F or dF would be in this case.