Numerical Methods for Hamilton-Jacobi-Bellman Equations

Abstract

In this work we considered HJB equations, that arise from stochastic optimal control problems with a finite time interval. If the diffusion is allowed to become degenerate, the solution cannot be understood in the classical sense. Therefore one needs the notion of viscosity solutions. With some stability and consistency assumptions, monotone methods provide the convergence to the viscosity solution. In this thesis we looked at monotone finite difference methods, semi lagragian methods and finite element methods for isotropic diffusion. In the last chapter we introduce the vanishing moment method, a method not based on monotonicity.