This problem is given as a motivating example for the Ito Integral in Mikosch's Elementary Stochastic Calculus with Finance in View...
Consider the Riemann-Stieltjes Sum $$S_n = \sum^n_{i=1} B_{t_{i-1}}\Delta_i B$$ where $$B=(B_t,t\geq 0)$$ is a Brownian Motion, $$\tau_n: 0=t_0<t_1<...<t_{n-1}<t_n=t$$ is a partition of $[0,t]$ and, for any function $f$ on $[0,t]$, $$\Delta f: \Delta_i f = f(t_i)-f(t_{i-1}), i=1,...,n,$$ are the corresponding increments of $f$ and $$\Delta_i = t_i - t_{i-1}, i=1,...,n.$$ The Riemann-Stieltjes sum $S_n$corresponds t the partition $\tau_n$ and the intermediate partition $(y_i)$ with $y_i=t_{i-1}$ is the left end point of the interval $[t_{i-1},t_i]$. It follows that $S_n$ can be written in the form $$S_n=\frac{1}{2}B^2_t - \frac{1}{2}\sum^n_{i=1}(\Delta_i B)^2.$$
I don't understand how the last equation for $S_n$ is derived since what I got is $$S_n=\frac{1}{2}B^2_{t_0}+B^2_{t_1}+B^2_{t_2}+...+B^{2}_{t_{n-1}}+\frac{1}{2}B^2_{t_n} - \frac{1}{2}\sum^n_{i=1}(\Delta_i B)^2$$ $$\Leftrightarrow S_n=B^2_{t_1}+B^2_{t_2}+...+B^{2}_{t_{n-1}}+\frac{1}{2}B^2_{t_n} - \frac{1}{2}\sum^n_{i=1}(\Delta_i B)^2.$$ My question is, where did $$B^2_{t_1}+B^2_{t_2}+...+B^{2}_{t_{n-1}}$$ go?
