I have the following problem. Let $H_t$ be an adapted process with trajectories a.s. of class $C^1$ on $\mathbb{R}_{+}$. Compute using simple process $\int_o^t H_s d B_s$.
My idea is to firstly set $U_n(s):=\sum_{i=0}^{m(n)-1} H_{t_i^n} \mathbb{1}_{(t_i^n-t_{i+1}^n]}$ (where of course I considered a partition of $[0,t]$) and show that this sequence of simple processes tends to $H_s$. This allows me to say that the sequence of the integrals of $U_n$ converges to the integral of $H_s$. Then, computing the integral of $U_n$ (as n $\to \infty$) we conclude.
Now, in showing that $U_n \to H_s$ I am note sure whether I can state the following : since $H_t$ is a.s. of class $C_1$ on $\mathbb{R}_{+}$, then
$|H_{t_i^n}-H_s| \leq M |t_i^n-s|$
for some constant $M$. I would say it holds thanks to Lagrange's theorem ($\omega$ per $\omega$) but Lagrange requires a function to be continuos on a closed set and differentiable on the open.
Thank you very much for any help :)