A gradient method for fully nonlinear parabolic differential equations with convex nonlinearity

We introduce a generic numerical scheme for fully nonlinear parabolic PDEs on the full domain, where the nonlinearity is convex on the Hessian of the solution. The main idea stems from reduction of a fully nonlinear problem to a class of simpler semilinear ones parameterized by the diffusion term. The contribution of this paper is to provide a directional maximum principle with respect to the diffusion coefficient for semilinear problems, which specifies how to modify the diffusion coefficient to approach the solution of the fully nonlinear problem. Although the diffusion coefficient is infinite-dimensional, the maximum direction of increase can be found explicitly. This also provides a numerical gradient method for the fully nonlinear problem. To establish a proof-of-concept, we test our method in a numerical experiment on the fully nonlinear Hamilton-Jacobi-Bellman equation for portfolio optimization in a stochastic volatility model.