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Let's assume $F(t,W(t))=t*W(t)$. ($W(t)$ is brownian motion)
Then when we calculate partial derivative of $F$ with $t$ (That is $F_t$),
$F_t=W(t)$
But I'm confused because the meaning of partial derivative is observe very small difference when one variable is changed very small with the other variable remaining constant.

So considering this meaning, when the variable $t$ has changed, then there are no way $W(t)$ remains constant!

Although we should make $W(t)$ constant, the $W(t)$ naturally dependent on variable $t$, So it cannot be constant intrinsically.

I have already seen related question : partial derivative
However, it said it depends on situations, when we write the function in the form $W$ or $W(t)$

But, I have never seen that the outcome is different depending on the author's style "$W$ or $W(t)$"

Then My Question : What is the basis that supports the logic, $F_t = W(t)$, in above equation?

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  • $\begingroup$ BM paths are nowhere differentiable, so the partial dertiavtive does not exist. $\endgroup$ – Kavi Rama Murthy Apr 1 at 5:02
  • $\begingroup$ @KaviRamaMurthy Yes you're right I already heard of it But, what I confused is when we apply ito's formula there is partial derivative term : $f_tdt$. Isn't it partial derivative? $\endgroup$ – user13232877 Apr 1 at 5:11

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