TITLE: Why do people write stochastic differential equations in differential form?
QUESTION [14 upvotes]: I am trying to teach myself about stochastic differential equations.  In several accounts I've read, the author defines an SDE as an integral equation, in which at least one integral is a stochastic integral, then  writes that in practice, people usually write the SDE in differential form.  My question is, why? 
There are two possible reasons I can guess: 
(i) If you write an SDE in differential form, it makes it easy to simulate sample paths using random walks, and you can think of the SDE as a limit of random walks.  This may provide some heuristic understanding that the integral form does not.
(ii) If you write an SDE in differential form, you can do "formal" computations with the differentials that are not necessarily rigourous, but lead to correct conclusions.  Personally, I have always had trouble understanding differential forms, and from what I've read, it is difficult to define exactly what $dW_t$ means (the differential of the Wiener process, i.e. Brownian motion).  I'm not even sure if $dW_t$ has a precise meaning.

REPLY [2 votes]: Intuitively, the symbol $\text{d}W_{t}$ may be interpreted as an infinitesimal increment of a (one-dimensional) Wiener process, $W$. $$\text{d}W_{t} = W_{t+\text{d}t}-W_{t}\ .$$ Increments of a Wiener process are normal distributed random variables whose expected value is zero and whose variance is equal to the time-increment. Thus the symbol $\text{d}W_{t}$ can be interpreted as a normal distributed random variable whose expected value is $0$ and whose variance is equal to $\text{d}t$. This implies that it is quite likely (probability close to 0.7) to find the random variable $\text{d}W_{t}$ located somewhere between the numbers $-\sqrt{\text{d}t}$ and $\sqrt{\text{d}t}\ .$ 
As to a precise meaning of the symbol $\text{d}W_{t}\ ,$ you have to be more specific what you mean by "precise". After all, what is the precise meaning of the symbol $\text{d}t$? (Without using 1-forms or nonstandard analysis.)