Demonstration with Itô's lemma

Question

This is my first question here, and I'm asking because I start to have doubts about whether the exercise is right or there are missing elements, and I wanted to ask before going to my professor. Do excuse me it it's been posted before, I've been looking for it for three days but maybe I didn't use the right terms. I have to solve the following exercise: Using Itô's lemma, demonstrate that if $f:(0, +\infty) \to \mathbb{R}$ is a function with a continuous derivative, then: $$\int^t_0 h_s dW_s = h_tW_t - \int^t_0 h_s' W_s ds$$ I understand the first term, $h_tW_t$ as $F_{W_t}$ in the usual Itô's lemma, $$\int^t_0 f_{W_t} dW_t = F_{W_t} - \frac 12 \int^t_0 f_{W_s}'ds,$$ but I can't see how $$\int^t_0 h_s' W_s ds = \frac 12 \int^t_0 f_{W_s}'ds$$ I have also doubts about wrt what is $h'$ derived. I'm also not fully sure how is $F_t$ $h_tW_t$ ie, why does $h_s$ go to $h_t$ if it's not integrated wrt $s$. I hope my LaTeX is going to come through as code, I'm sorry if it doesn't. Thank you for any help.

Answer 1

Note: I'll use $W(t)$ notation here instead of $W_t$.

Observe that the given statement is equivalent to the following differential form: $$ {\rm d}(h(t) W(t)) = h(t) {\rm d} W(t) + h'(t)W(t) {\rm d}t. $$ Your choice of $F$ to apply Itô's lemma is correct: it is $F(t, W(t))$, where $F(t, x) = h(t)\cdot x$. There is a special case of the lemma for transformations of Wiener process: $$ {\rm d} F(t, W(t)) = F'_t(t, W(t)) {\rm d} t + F'_x(t,W(t)) {\rm d}W(t) + \frac{1}{2}F''_{xx}(t, W(t)) {\rm d}t. $$ As $F'_t(t,x) = h'(t)\cdot x$, $F'_x(t,x) = h(t)$ and $F''_{xx}(t,x) = 0$, it follows that $$ {\rm d} F(t, W(t)) = h'(t) W(t) {\rm d}t + h'(t)W(t) {\rm d}t. $$