Just a few notes
How to make sense of $\text dW_t$ is the entire point of stochastic calculus. It's far beyond the scope of any answer here. You should read some introductory lecture notes/books on stochastic calculus. You could start here.
- The idea: Riemann-Stieltjes integrals are of the form $\int_0^t f(s)\mathrm{d}g(s)$ and are well-defined if $f$ is continuous and $g$ has bounded variation, see also this answer. Brownian motion does not have finite variation. But Brownian motion has finite quadratic variation. We thus define a new integral, $I_t=\int_0^t X_s\text{d}W_s$ which converges in a (weaker) mean squared ($L^2$) sense. The construction is still the same: define this integral for step functions (which take random values over certain intervals) and approximate any well-behaved process $X_t$ by these step functions. The result is the Itô integral. One key property is that it is a martingale (e.g. $\text{d}I_t=X_t\text{d}W_t$ is driftless). I omitted many technicalities of course.
In the simplest case, the function $f$ needs to be smooth. Weaker conditions are possible, see this answer. You can take functions like $f(x)=x^2$, $f(t,x)=tx$ or indeed $f(t,x_1,...,x_n)$. These are ``standard'' functions. You then consider processes like $f(X_t)=X_t^2$ or $f(X_t)=tX_t$ by mechanically plugging in the process $X_t$ for the variable $x$.
- It's a bit like algebra and polynomials: You have some general rule $p(X)=X+X^2$ and you can plug in elements from your ring/field (numbers) or for example fancier objects such as matrices and other linear maps.
- The entire point of Itô's Lemma is that if you know the process $X_t$ but are interested in a process $f(X_t)$: for example, you have a model for variances $v_t$ but you're interested in volatilities $\sqrt{v_t}$ or you know a model for the stock price $S_t$ but are interested in the dynamics of futures prices. Itô's Lemma is thus some stochastic version of the chain rule.
$\text dX_t^2\neq \text dt$. Instead, $\text dW_t^2=\text dt$ and $\text dX_t^2 = \sigma^2(t,X_t)\text dt$
Derivatives like $W'(t)=\lim\limits_{h\to0}\frac{W_{t+h}-W_t}{h}$ do not exist, see here. Sample paths of Brownian motion are continuous but nowhere differentiable. Something like $\frac{\partial}{\partial W_t}$ does not make sense. In fact, the term ``$\text{d}W_t$'' technically doesn't make sense as a differential and is just shorthand notation for an integral, $\text{d}X_t=\sigma_t\text{d}W_t$ really only means $X_t=X_0+\int_0^t\sigma_s\text{d}W_s$. The differential notation is just shorter and handier.
Heuristic proof for Itô's Lemma
Consider a function $f(t,x)$ and an Itô process $\text{d}X_t=\mu(t,X_t)\text{d}t+\sigma(t,X_t)\text{d}W_t$. Taylor tells us
\begin{align*}
\text df(t,x) = f_t(t,x)\text dt+f_x(t,x)\text dx+\frac{1}{2}f_{tt}(t,x)\text dt^2+f_{tx}(t,x)\text dx\text dt+\frac{1}{2}f_{xx}(t,x)\text dx^2,
\end{align*}
where subscripts refer to partial derivatives. Now, we plug in mechanically $X_t$ for $x$ and obtain
\begin{align*}
\text df(t,X_t) = f_t(t,X_t)\text dt+f_x(t,X_t)\text dX_t+\frac{1}{2}f_{tt}(t,X_t)\text dt^2+f_{tx}(t,X_t)\text dX_t\text dt+\frac{1}{2}f_{xx}(t,X_t)\text dX_t^2
\end{align*}
As $\text dt\to0$, we can ignore $\text dt^2$. In terms of magnitude, $\text{d}X_t\sim\sqrt{\text{d}t}$ and $\text{d}X_t^2\sim\text{d}t$. We can thus ignore $\text dX_t\text dt\sim \text{d}t^{3/2}$ but we cannot ignore $\text dX_t^2$ which is of order $\text{d}t$! This is the big difference for stochastic calculus from ordinary real calculus for which we can ignore such terms. Thus,
\begin{align*}
\text df(t,X_t) &= f_t(t,X_t)\text dt+f_x(t,x)\text dX_t+\frac{1}{2}f_{xx}(t,X_t)\sigma^2(t,X_t)\text dt \\
&= \left( f_t(t,X_t)+f_x(t,X_t)\mu(t,X_t)+\frac{1}{2}f_{xx}(t,X_t)\sigma^2(t,X_t)\right)\text{d}t+f_x(t,X_t)\sigma(t,X_t)\text{d}W_t,
\end{align*}
which is the standard formula you see in textbooks and on wikipedia.
Example for Itô's Lemma
We want to compute $\int_0^t W_s\text{d}W_s$. As it turns out, a clever way is to study $f(t,x)=x^2$ with $\mu(t,X_t)=0$ and $\sigma(t,X_t)=1$, i.e. $X_t=W_t$ is a standard Brownian motion. Then,
\begin{align*}
\text dW_t^2&=\left(0+0+\frac{1}{2}\cdot1\cdot2\right)\text{d}t+2W_t\text{d}W_t \\
\implies \int_0^t W_s\text{d}W_s&=\frac{1}{2}W_t^2-\frac{1}{2}t
\end{align*}
The key difference to ``ordinary'' calculus, i.e. $\int x\text{d}x=\frac{1}{2}x^2$ is the term $-\frac{1}{2}t$ in the Itô integral. It comes from the mere fact that you can't ignore terms like $\text{d}X_t^2$ for stochastic processes (which have non-zero quadratic variation). In fact, it stems from the $\frac{1}{2}f_{xx}(t,X_t)\sigma^2(t,X_t)\text{d}t$ part.
Plugging in $X_t$ for $x$
This point is simple yet subtle. It's mainly due to notation. Consider $f(x)=x^2$. This function takes some input ($x$) and gives you some output ($x^2$). You can substitute anything for the variable (placeholder) $x$ for which you can define powers. For example,
- if $(a_n)$ is a sequence of real numbers, then $f(a_n)=a_n^2$ is a new sequence of numbers
- if $x$ is a real number, then $f(x)=x^2$ is another real number
- if $A\in K^{n\times n}$ is a square matrix, then $f(A)=A^2$ is another square matrix
- if $(X_t)_{t\geq0}$ is a stochastic process, then $f(X_t)=X_t^2$ is another stochastic process
Suppose $r_t$ is a process for the short rate. For example, Vasicek proposes $\text{d}r_t=\kappa(\theta-r_t)\text{d}t+\sigma\text{d}W_t$. The price of a zero-coupon bond is $e^{A(\tau)+r_tB(\tau)}$ for some functions $A,B$. You could now be interested in knowing the dynamics of the bond price, $\text{d}P$. You would thus use the function $f(t,x)=e^{A+xB}$ which, when you plug in $r_t$ for $x$ gives you the bond price.
It's confusing because it's often convenient to be a bit sloppy with notation. You often see the Black-Scholes solution being written as $V(t,S_t)=S_t\Phi(d_1)-Ke^{-rT}\Phi(d_2)$ where $$\frac{\partial V}{\partial t}+(r-\delta) S_t\frac{\partial V}{\partial S_t}+\frac{1}{2}\sigma^2S_t^2\frac{\partial^2 V}{\partial S_t^2}-rV=0$$ which is however nonsense. You should technically write something along the lines of the call option price is $V(t,S_t)$ where $V(t,x)=x\Phi(d_1)+Ke^{-rT}\Phi(d_2)$. The function $V$ satisfies $$\frac{\partial V}{\partial t}+(r-\delta) x\frac{\partial V}{\partial x}+\frac{1}{2}\sigma^2x^2\frac{\partial^2 V}{\partial x^2}-rV=0.$$ The difference is that $V(t,x)$ is a ``normal'' function which you can differentiate with respect to $x$. An expression like $\frac{\partial V}{\partial S_t}$ doesn't make any sense. Often, it's convenient to use this shorthand notation if your audience knows that you mean but it must be terribly confusing for students starting to learn about finance.
When deriving Itô's Lemma, you start with the Taylor expansion of the function $f(t,x)$. At this stage, $f$ is an arbitrary (real-valued) function. After computing the partial derivatives of $f$, you then simply plug in the stochastic process $X_t$ for the variable $x$. Remember: the variable $x$ is just a placeholder for something else (in our case: a stochastic process).
