Using a martingale property to deduce orthogonality of some increments

I know that if $(M_t)_{t\ge0}$ is a continuous Martingale with $EM_t^2<\infty$ for every $t\ge0$, then its increments are orthogonal. Is there a way to deduce from this result directly that $g(B_{t_{i-1}})\left\{(B_{t_{i}}-B_{t_{i-1}})^2-(t_i-t_{i-1})\right\}$ are orthogonal for different $i$'s, where $B_t$ is a Brownian motion on $[0,T]$ and $0=t_1<\ldots<t_n=T$ is a partition of the interval and $f$ is continuous and bounded. I know that $B_t^2-t$ is a Martingale but its increments are $B_{t_i}^2-t_i-(B_{t_{i-1}})-t_{i-1}$ which it seems I cannot directly apply...

Solution 1:

Let $\mathcal F_i$ be the $\sigma$-algebra generated by the random variables $B_{t_k}$, $0\leqslant k\leqslant i$. Since $g\left(B_{t_{i-1}}\right)$ is $\mathcal F_{i-1}$-measurable, we derive that $$ \mathbb E\left[g\left(B_{t_{i-1}}\right)\left\{(B_{t_{i}}-B_{t_{i-1}})^2-(t_i-t_{i-1})\right\}\mid \mathcal F_{i-1}\right]=g(B_{t_{i-1}}) \mathbb E\left[\left\{(B_{t_{i}}-B_{t_{i-1}})^2-(t_i-t_{i-1})\right\}\mid \mathcal F_{i-1}\right]. $$ Moreover, the random variable $B_{t_{i}}-B_{t_{i-1}}$ is independent of $\mathcal F_{i-1}$ hence so is $\left(B_{t_{i}}-B_{t_{i-1}}\right)^2-(t_i-t_{i-1})$ and we get that

$$ \mathbb E\left[g\left(B_{t_{i-1}}\right)\left\{(B_{t_{i}}-B_{t_{i-1}})^2-(t_i-t_{i-1})\right\}\mid \mathcal F_{i-1}\right]=0.$$