  In this paper, we propose a monotone mixed finite difference scheme for
solving the two-dimensional Monge-Amp\`ere equation. In order to accomplish
this, we convert the Monge-Amp\`ere equation to an equivalent
Hamilton-Jacobi-Bellman (HJB) equation. Based on the HJB formulation, we apply
the standard 7-point stencil discretization, which is second order accurate, to
the grid points wherever monotonicity holds, and apply semi-Lagrangian wide
stencil discretization elsewhere to ensure monotonicity on the entire
computational domain. By dividing the admissible control set into six regions
and optimizing the sub-problem in each region, the computational cost of the
optimization problem at each grid point is reduced from $O(M^2)$ to $O(1)$ when
the standard 7-point stencil discretization is applied and to $O(M)$ otherwise,
where the discretized control set is $M \times M$. We prove that our numerical
scheme satisfies consistency, stability, monotonicity and strong comparison
principle, and hence is convergent to the viscosity solution of the
Monge-Amp\`ere equation. In the numerical results, second order convergence
rate is achieved when the standard 7-point stencil discretization is applied
monotonically on the entire computation domain, and up to order one convergence
is achieved otherwise. The proposed mixed scheme yields a smaller
discretization error and a faster convergence rate compared to the pure
semi-Lagrangian wide stencil scheme.
