Generalization of Ito's Lemma to composite function

Question

Ito's Lemma gives that for a function $F$ of a stochastic variable $X$, $dF = \frac{dF}{dX}dX + \frac{1}{2}\frac{d^2F}{dX^2}dt$

Given a stochastic differential equation $dS = a(S) dt + b(S) dX$ and a function $V(S)$, the differential equation satisfied by $V(S)$ is stated to be $dV = \frac{dV}{dS}dS + \frac{1}{2}b^2 \frac{d^2V}{dS^2}dt $. The book says this can be derived properly or the cheating substitution of $dX^2 = dt$ could be used.

But I just don't understand how the expression for $dV$ is obtained - either properly or by using the Taylor series. While using the Taylor series, how should I expand $V$? Should I use the Taylor series for a composite function, as shown, for instance, here? And how might I do it "properly"?

The book is Quantitative Finance of Paul Wilmott, I have made a screenshot of the relevant section.

I make a completely naive attempt by trying to apply Ito's lemma to $V$ treating $S$ as the random variable (since it is, in effect the function of one).

$dV = \frac{dV}{dS}dS + \frac{1}{2}\frac{d^2V}{dS^2}dS^2 $

and then substituting the given expression of $dS$ in the second term of the RHS. $dV = \frac{dV}{dS}dS + \frac{1}{2}\frac{d^2V}{dS^2}(a(S) dt + b(S) dX)^2$

This can directly give the correct expression for $dV$ if the $dt$ term inside the braces can be made to go to zero. But why would that be? So obviously there's something subtle going on that I am completely missing. Any pointers would be appreciated. Also regarding how to derive it properly.

Answer 1

In analysis, you generally consider (on the infinitesimal scale) only first-order variation ($dt$, $dx$) since continuously differentiable functions have bounded variation on any finite interval, hence zero quadratic variation: " $\left(dt\right)^2 = 0$".

However, this is not the case of Brownian motion, which has infinite variation on any finite interval; yet, it has finite quadratic variation, $d\langle W\rangle_t = dt$. This is why you have to consider its second-order variation, or quadratic variation (see Quadratic Variation on Wikipedia). You also have $d\langle t, W_t\rangle \approx \left(dt\right)\left(dW_t\right)$, which has zero mean and a negligible variance of order $\left(dt\right)^3$.

At the end of the day, the only nonnegligible term in the $\left(dS_t\right)^2$ term is the $b\left(S\right)^2 dt$ term.

Paul Wilmott's books are absolutely great, but they are definitely not the right ones to take for mathematical rigor ;) if you want a good finance textbook with a fair stochastic analysis introduction, I recommend Martingale Methods in Financial Modelling from Musiela and Rutkowski.