of $f[m](x_{i,j})$ is given by $(f_h[M])_{i,j} = f[m_h](x_{i,j})$. In particular, if $f$ is a local operator, i.e. $f[m](x)=f(m(x))$, then we set $f_h[M]_{j,k}=f(M_{j,k})$.\
We assume that:

-   $f_h$ is continuous and maps ${\mathcal{K}}_h$ on a bounded set of grid functions.

-   $f_h$ is monotone, i.e. for $M,\bar M\in {\mathcal{K}}_h$ $$\left( f_h[M]-f_h[\bar M],M-\bar  M \right)_2 \le 0 \Rightarrow  M=  \bar M.$$

-   There exists $C$ independent of $h$ such that for all grid functions $M\in {\mathcal{K}}_h$ $$\begin{aligned}
        & \| f_h[M] \|_{\infty} \le C\\
        & | f_h[M]_{i,j} - f_h[M]_{\bar i,\bar j} | \le C |x_{i,j}-x_{\bar i,\bar j}|\qquad \forall 0\le i,j,\bar i,\bar j \le N_h.
        \end{aligned}$$

-   For any $m\in {\mathcal{K}}=\{m\in L^1({\mathbb{T}}^2): m\ge 0, \int_{{\mathbb{T}}^2}m(x)dx=1\}$, defined $$M_h=\frac 1 {h^2} \int_{|x-x_{i,j}|_{\infty}\le h/2} m(x)dx,$$ then $$\lim_{h\to 0}\,\sup_{m\in{\mathcal{K}}}\|f[m]-f_h[M_h]\|_\infty =0.$$

We approximate the Hamilton-Jacobi-Bellman equation in by means of the scheme $$\label{discrete_HJequation}
\left\{
\begin{array}{ll}
\bar D_{\Delta t}^{\alpha} U^n_{i,j} - (\Delta_h U^{n})_{i,j} + g(x_{i,j},[D_h U^{n}]_{i,j}) = f_h[M^{n+1}]_{i,j}, \quad& n=0,\dots, N-1,\, 0\le i,j\le N_h,\\[6pt]
U^N_{i,j}=u_T(x_{i,j})& 0\le i,j\le N_h,
\end{array}
\right.$$ for a given $M= (M^n)_{ 0\le n\le N-1}$. Note that the previous scheme is backward, implicit and non local in the time variable, since the solution at step $n$ depends on the values of the solution computed at previous steps 