*2.5. Proposition*

In the above fractional differencing scheme, the memory length *L* is variable and is a fixed number in each differencing calculation. This is particularly important for efficient simulation. Otherwise, the memory length *t* − *a* will be a function of the variable *t*. According to Equations (18) and (24), when *L* is large, the calculation time with difference scheme (24) will be less than that of difference scheme of (18). However, as the value of *L* increases, its influence on the derivative will decrease. Therefore, proper choice of the memory length *L* will balance the accuracy of calculation and the time cost. In addition, the fractional differencing scheme (24) is stable, since the residual *Rih* related to discrete step *h* is bounded by the infinitesimal equivalent value of *h*2, and the fact that for any 1 <α< 2,

$$\begin{split} \mathbf{a}^{\mathbf{a}} (\mathbf{x} + 1)^{\mathbf{a}} - \mathbf{x}^{\mathbf{a}} &= \sum\_{k=1}^{\infty} \frac{a(a-1)\cdots(a-k+1)}{k!} \frac{1}{\mathbf{x}^{k-a}} \\ &= a \frac{1}{\mathbf{x}^{1-a}} + \frac{a(a-1)}{2} \frac{1}{\mathbf{x}^{2-a}} + \sum\_{k=3}^{\infty} \frac{a(a-1)\cdots(a-k+1)}{k!} \frac{1}{\mathbf{x}^{k-a}} . \end{split} \tag{26}$$

The above expression is bounded. In next section, we will show how these values of memory length influence the results.
