= Brownian weighted quadratic variation in L2
{c}
{title2=$\sum_jB_{t_j}(B_{t_{j+1}}-B_{t_j})^2\longrightarrow\int_0^tB_sds$}
For <Brownian motion> and deterministic <partitions of an interval> $[0,t]$ whose mesh $|\pi|$ tends to zero, the displayed limit holds in <L2 space>. Split the error into $U_\pi=\sum_jB_{t_j}((\Delta B_j)^2-\Delta t_j)$ and the <Riemann sum> error $V_\pi=\sum_jB_{t_j}\Delta t_j-\int_0^tB_sds$. The summands of $U_\pi$ are <martingale differences>, hence <orthogonal> in <L2 space>, and the <Gaussian fourth moment> gives
$$
\mathbb EU_\pi^2=2\sum_jt_j(\Delta t_j)^2\leq2t^2|\pi|.
$$
The <Cauchy-Schwarz inequality> and the <Brownian increment> <variance> give $\mathbb EV_\pi^2\leq(t/2)\sum_j(\Delta t_j)^2\leq t^2|\pi|/2$. Both vanish. This proves the limit directly without a stochastic change-of-variables formula.
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