Let \(B\) be the standard Brownian motion and \(B^H\) fractional Brownian motion with Hurst index \(H\in (0,1]\). If the Brownian motion \(B\) and the fractional Brownian motion \(B^H\) are independent and \(\alpha\in\mathbb{R} \setminus \{0\}\), define the mixed fractional Brownian motion \(M^{H,\alpha}\) by \(M^{H,\alpha} \doteq B+\alpha B^H\). If \(X\) is a stochastic process, denote by \(\mathbb{F}^X\) the filtration generated by \(X\). If \(f=\sum^{n-1}_{j=0} f_jI_{(t_j, t_{j+1}]}\) is a simple \(\mathbb{F}^X\)-predictable process on \((0,1]\): \(|f|\leq 1\), \(f_j\in F_{t_j}^X\) and \(0\leq t_0