This is from the book on Brownian Motion by Morters and Peres.
let $(B_t)_{t\in [0,1]}$ be a standard Brownian motion and $F\in C[0,1]$ differentiable. Then define
$$\xi_n=2^n\sum_{j=1}^{2^n}\left[F\left(\frac{j}{2^n}\right)-F\left(\frac{j-1}{2^n}\right)\right]\left[B\left(\frac{j}{2^n}\right)-B\left(\frac{j-1}{2^n}\right)\right]$$
and it is claimed that
$$\xi_n-\xi_{n-1}=2^n\sigma_n\sum_{j=1}^{2^{n-1}}\left[2F\left(\frac{2j-1}{2^n}\right)-F\left(\frac{2j-2}{2^n}\right)-F\left(\frac{2j}{2^n}\right)\right]Z\left(\frac{2j-1}{2^n}\right)$$
where $\sigma_n = 2^{-(n+1)/2}$. And it is suggested that the identity follows naturally from Levy's construction of Brownian Motion
$$B\left(\frac{2j-1}{2^n}\right)=\frac{1}{2}\left[B\left(\frac{(2j-2)}{2^n}\right)+B\left(\frac{2j}{2^n}\right)\right]+\sigma_n Z\left(\frac{2j-1}{2^n}\right)$$ where $Z(t)$ are i.i.d standard normal R.V.s.
How can it be proved?