Use Ito's Lemma to show:

Question

I am somewhat unsure how to go about showing this:

Use Ito's Lemma to show for any deterministic differentiable function, $f$: $$ \int_0^t f(s) dB(s) = f(t)B(t) - \int_0^t B(s)f'(s)ds $$ Where $B(t)$ is Brownian motion

I've used Ito's Lemma to show things are martingales, but I'm not sure how to go about doing this. I'm guessing it's something to do with the fact that f is deterministic, and Brownian motion is random, but am not really sure.

Cheers.

EDIT: Ito's lemma, as defined in my notes in integral form:

For $X_t = h(t, B(t))$ $$ X_t = X_0 + \int_0^th_B(s, B(s))dB(s) + \int_0^t(h_t(s, B(s)) + \frac{1}{2}h_{BB}(s, B(s))ds $$

Answer 1

By your definition $h(t,b) = f(t)b$ . Just plug right in. Basically the $X_0$ term goes away because $B_0 =0$ and the second derivative term goes away because h is linear in b. Then it's just a matter of isolating the Ito integral on the left hand side.