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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.

1 Answers 1

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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.

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    Do you know if/why my answer is incorrect (if yours is correct)? I know that it's false to say that "you can't write partial derivatives w.r.t. BM". For example apply Ito's lemma to the function $f(t,x)$, where $x$ is $B(t)$ (in fact this is one of the most standard routines in stochastic finance). Applying Ito's lemma you will end up having to evaluate $\frac{\partial f}{\partial x}(t,x)$.2012-12-04