Finite element methods for Bellman and Isaacs Equations

This work concerns the numerical analysis of the Partial Differential Equations (PDEs) with a particular focus on fully nonlinear PDEs. More specifically, the main goal is to provide a finite element method to approximate solutions of Isaacs equations, which come from game theory and can be thought of as generalisation of Hamilton-Jacobi-Bellman (HJB) equations. Both of these classes of problems arise from the stochastic optimal control problems. Is is widely known that nonlinear PDEs do not in general admit classical solutions. A way to circumvent this issue is to use a relaxed definition of derivative leading to the notion of a generalised solution. One such notion is that of viscosity solution introduced in 1980s by Crandall and Lions. The main idea is to regularise non-smooth functions by using comparison principles and subtractive testing. The theory of viscosity solutions gave rise to novel numerical methods. A general framework of formulating convergent numerical schemes for (possibly degenerate) elliptic PDEs was formulated by Barles and Souganidis in 1991. The main result states that, given a comparison principle depending on the application at hand, a monotone, stable and consistent numerical scheme converges to the unique viscosity solution of a fully nonlinear problem. This framework is used throughout this work to formulate convergent numerical schemes. The main three contributions of the thesis are as follows. First we present a Finite Element Method to approximate solutions of isotropic parabolic problems of Isaacs type with possibly degenerate diffusions. Second we design a method of numerically approximating isotropic parabolic Hamilton-Jacobi-Bellman equations with nonlinear, mixed boundary conditions where Robin type boundary conditions are imposed via one-sided Dini derivatives. In both cases we prove the convergence of the numerical solution to the unique viscosity solution. The uniqueness of numerical solution is guaranteed by Howard’s algorithm. The analysis of the HJB equations with mixed boundary conditions is motivated by option pricing in a financial setting, which leads to our third contribution. We extend the Heston model of mathematical finance to permit the uncertain market price of volatility risk and we interpret it as an HJB equation. Finally, we present a case study investigating the effects of the market price of volatility risk on the option value and its derivatives.