TITLE: Itô's formula: Differential form
QUESTION [12 upvotes]: I've started a course on financial mathematics and I'm currently being introduced to stochastical analysis, spesifically Itô's formula. From the book:

It is sometimes useful to use the following shorthand version of [Itô's formula]: $$ \mathrm{d}f(t,X(t)) = \dfrac{\partial f(t,X(t))}{\partial t}\mathrm{d}t + \dfrac{\partial f(t,X(t))}{\partial x}\mathrm{d}X(t)
+\dfrac{1}{2}\dfrac{\partial^2 f(t,X(t))}{\partial x^2}(\mathrm{d}X(t))^2  $$ together with the calculation rules $ (\mathrm{d} t)^2 =0,\ \mathrm{d}t\mathrm{d}B(t) = \mathrm{d}B(t)\mathrm{d}t = 0 \text{ and } (\mathrm{d}B(t))^2 = \mathrm{d}t$

$f(t,x)$ is a function twice differentiable in $t$ and twice in $x$. $X(t)$ is a semimartingale. The definition of a semimartingale used in the book is (in case there is several definitions)

The stochastic process $X(t)$ is a semimartingale if there exists two Itô integrable stochastic processes $Y(t)$ and $Z(t)$ such that 
  $$ X(t) = x + \int_0^t Y(s) \mathrm{d}B(s) + \int_0^t Z(s)\mathrm{d}s $$

I'm having a hard time with understanding this  short version of Itô's formula. The reason for this is that I'm not familiar with these differential-terms (name?) of type $\mathrm{d}t$ other than in the classical integral. These "calculation rules" that the author speaks of are entirely Greek to me. How should I think of these terms? 
Any written, preferably online, resources explaining these terms for a newcomer is also appreciated.

REPLY [4 votes]: A simplified way of thinking about this is to interpret the equation as saying what approximately happens to the value in the left hand side when time advances a little. 
If the present time is $t$ and a very short interval of time $\Delta t$ passes, then the value of $f(t,X(t))$ will change approximately by the amount
$$ \dfrac{\partial f(t,X(t))}{\partial t}\Delta t + \dfrac{\partial f(t,X(t))}{\partial x}\Delta X(t)
+\dfrac{1}{2}\dfrac{\partial^2 f(t,X(t))}{\partial x^2}(\Delta X(t))^2.  $$
Note, that this approximation involves approximating $\Delta X(t)$, i.e. how the process $X$ changes between $t$ and $t+\Delta t$. This depends on how the process $X$ is defined. 
If the process $X$ is defined in terms of time and a Brownian process, for example
\begin{align} \mathrm dX(t) &= \mathrm d t + \sigma \mathrm d B(t) \\
\overset{\int_0^t} \implies X(t) - X(0) &= t + \sigma B(t),\end{align}
then $\Delta X(t)$ will depend on $\Delta t$ and $\Delta B(t)$, so you will encounter products of the form $\Delta t \Delta B(t), \Delta t ^ 2$ and $\Delta B(t)^2$ while trying to find $\Delta X(t)^2$. 
The expression $\Delta t \Delta B = 0$ tells you to discard any terms containing $\Delta t \Delta B$ in your calculation of $\Delta X(t)$. Similarly, the other relations tell you how to get rid of the other "products of differentials". 
The mathematical justification of this is that in the limit when $\Delta t \to 0$ (which you take when integrating) those products behave just like their simpler alternative forms. That is, $\Delta t^2$ and $\Delta t \Delta B$ become negligible and, surprisingly, $\Delta B^2$ behaves just like $\Delta t$.