Gradient ascent method for fully nonlinear parabolic differential equations with convex nonlinearity

TL;DR

This paper presents a numerical gradient ascent method for solving fully nonlinear parabolic PDEs with convex nonlinearity by reducing them to semilinear problems and applying a directional maximum principle, demonstrated on a portfolio optimization problem.

Contribution

It introduces a novel approach to solve fully nonlinear PDEs by linking them to semilinear problems and providing a maximum principle for the diffusion coefficient, enabling a gradient ascent algorithm.

Findings

Effective numerical scheme for fully nonlinear PDEs.

Successful application to Hamilton-Jacobi-Bellman equation.

Explicit computation of the maximum direction for diffusion coefficient.

Abstract

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 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 to the solution of the fully nonlinear problem. While the objects of the study, diffusion coefficient, is infinite dimensional, the maximum direction of increase can be found explicitly. This also provides a numerical gradient ascent 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 under stochastic volatility model.