Partial Derivative of an Integral

Question

If $f(t)$ is a deterministic function of $t$ and $B_{n}$ is a brownian motion and:

$Z =\displaystyle\int^t_0 f(s)d\left(B(s)\right)$

How does one take the partial derivatives wrt to $t$ and $B_n$ on an integral like this?

I know $dZ = f(t)dB(t)$

Is this just?...

$\dfrac{\partial z}{\partial t} = f(t)$

and

$\dfrac{\partial z}{\partial B} = f(t)dB(t)$

Looking to apply the Ito formula on a bigger problem but stuck on this. Thanks.

Answer 1

I think Z(t) can not be differentiated with respect to time. All you can do is use Ito's lemma as you have already correctly done. Why I say the partial derivative does not exist is Brownian motion is not smooth and the paths of Brownian motion are almost surely non differentiable hence Z too is non differentiable.

http://www2.math.uu.se/~takis/L/BMseminar/BMnotes03_continuity.pdf

If you post the complete problem I might be able to help more.

regards.