# How to differentiate a brownian motion?

By definition a wiener process cannot be differentiated.

But when we use Ito's lemma on $F = X^2$, where X is wiener process

we have total change in

$$dF = 2XdX + dt$$

How can we calculate $\frac{dF}{dX}$ when by definition it cannot be differentiated? Isin't this contradiction by definition?

• Have you looked at this Wikipedia page? – Bob Jansen Apr 26 '15 at 9:14
• yes...wikipedia doesn't answer my question...see my comment below. Even for writing the integral form, how do I get dF/dX. – Animesh Saxena Apr 27 '15 at 11:11
• I think you are mixing things here. The function $F: x \mapsto x^2$ is differentiable but not the Wiener process itself :) – byouness May 15 '18 at 15:51

In order to apply Ito's lemma, your function needs to be a twice-differentiable function. There is no issue with the non-differentiability of the Wiener process. $\frac{dF}{dX}$ involves differentiating F, not the Wiener process X.
Using a simple analogy: instantaneous velocity ($\frac{dD}{dt}$) is the derivative of position (D) over time; what is differentiated is not time, but distance. I believe this is where your confusion stems from.
We write the differential form of Ito formula for simplification. Actually, the differential form for Ito formula $$dF(W(t)) = 2W(t)dW(t) + dt$$ means the integral form for Ito formula, $$\int{dF} = \int{2W(t)dW(t)} + \int{dt}$$ which make sense in mathemaitcs.
• We get $2W(t)$ by using Ito formula. And I think ocstl answered your question. – logistic Apr 27 '15 at 13:29