Ito's integral converge in probability https://en.wikipedia.org/wiki/Itô_calculus
From wikipedia:
$$\int_0^tH_tdB_t\equiv \lim_{n\rightarrow\infty}\sum_{i=1}^nH_{t_i}(B_{t_i}-B_{t_{i-1}})$$
It can be shown that this limit converges in probability. i.e.
$$P(\lim_{n\rightarrow\infty}\sum_{i=1}^nH_{t_i}(B_{t_i}-B_{t_{i-1}})\text{ exists})=1$$
Can someone give a proof or a reference how it converges in probability?
It is known that the brownian motion is almost surely of unbounded variation, but can we say the limit doesn't exists almost surely?

The definition of Ito's integral is based on the fact that $L^2$ is complete. So the limit you mentioned above is in the sense of $L^2$-convergence indeed. Consequently, $L^2$-convergence implies convergence in probability.
