  We introduce a generic numerical schemes for fully nonlinear parabolic PDEs
on the full domain, where the nonlinearity is convex on the Hessian of the
solution. The main idea behind this paper is reduction: we reduce the fully
nonlinear problem to a series of simpler semilinear ones parameterized by the
diffusion term and take the maximum over the solution of those problems through
a novel gradient ascend method on the diffusion coefficient that we introduced
for such semilinear PDEs. The promise of this study is a step toward the
proof-of-concept that for any arbitrary numerical scheme for semilinear
parabolic PDEs, one can build an iterative methods based on gradient ascent to
converge to the solution of a fully nonlinear problem. Therefore, investment in
our effort to create more efficient semilinear PDEs leads to efficiency for
fully nonlinear PDEs. We leverage the theoretical results of Soner et al.
[2012] to obtain numerical schemes for fully nonlinear PDEs and to justify the
convergence in our methodology. The method is tested in a numerical experiment
on the fully problem of portfolio optimization under stochastic volatility
model.
