$L^2$ convergence of Riemann integral of stochastic process

Question

I recently needed to verify that the left-hand Riemann sums, $R_N:=\sum_{i=1}^{N-1} W_{i/N}/N$ of a Brownian motion on $\mathbb{R}$ in the time interval $[0,1]$ converge to the Riemann integral $\int_{0}^1 W_t dt$ in the sense of $L^2(\Omega)$. To do this, I applied the dominated convergence theorem, using the fact that the supremum of Brownian motion in the interval $[0,1]$ is in $L^2(\Omega)$. Since I proved this last fact rather explicitely, I wondered if there is a more general argument, or if something like a $L^2(\Omega;\|\cdot\|_{L^{\infty}[0,1]})$ bound on the stochastic process is really necessary to ensure $L^2(\Omega)$ convergence of the left-hand Riemann sums.

Answer 1

Doob's and Burkholder's martingale inequalities can replace explicit calculations to show that the supremum over $t\in[0,1]$ of $|M_t|$ is finite, for general (sub)martingales (OP=special case $M_t=B_t$)