# relationship of SDE in Langevin equation form and Ito form

A formal SDE can be written in a way as (ito form):

$dx(t)=ax(t)dt+dw(t)$

where $w(t)$ is brownian motion.

Another way is to write the SDE (Langevin equation form) is

$\frac{dx(t)}{dt}=ax(t)+w(t)$

Can i ask whether there is any equivalence between these two forms?

## 1 Answer

The answer boils down, I think, to the fact that the Langevin form was developed before Ito Calculus and in the context of physics. In that case, some of the some of the mathematical subtleties can be waved away by pointing out that we are studying physical systems, so there's e.g. a minimum $\delta t$ which is too short for a fluid particle to hit the particle undergoing Brownian motion. (I'm not saying that's a good argument, since at the timescales anyone cares about, it's really better modeled as a Wiener process).

On the other hand, Wiener processes formalize the observations of Brownian motion and they have the property, among others, that independent increments are normally distributed with variance $\delta t$. This, eventually, leads to the fact that they are discontinuous with respect to time. In the Langevin formulation, one is implicitly assuming that $w(t) = dv(t)/dt$ for some well-behaved $v(t)$; setting $a = 0$ you see that the $v(t)$ is Brownian motion. But the fact that it's discontinuous as shown by Wiener means the $dt \rightarrow 0$ limit in the differential equation doesn't exist. Therefore, $dW$ in the Ito formulation is a formally correct way to write "$w(t) dt$" in a way such that the limits exist.