Numerical Methods for Hamilton-Jacobi-Bellman Equations in Finance

Project level: Honours, Masters, PhD

Many popular problems in financial mathematics can be posed in terms of a stochastic optimal control formulation, leading to the formulation of nonlinear Hamilton-Jacobi-Bellman (HJB) equations. The inherent challenges in solving these HJB equations include the lack of analytical solutions under realistic scenarios where controls are constrained, and the non-uniqueness and lack of smooth classical solutions due to their nonlinear nature. Consequently, our pursuit is directed towards finding the financially relevant solution for these HJB equations – the viscosity solution in this context.

A number of my projects are centered around the development of efficient numerical methods that ensure convergence to the viscosity solution for HJB equations arising in finance. Potential applications include portfolio optimisation (superannuation), variable annuities with riders (pension products), and valuation adjustments (regulations).

These projects, suitable for Honours, Master and PhD level students, emphasize the practical and real-world relevance of research in mathematical finance, offering opportunities for intellectual growth and for making meaningful contributions to understanding and controlling complex financial systems.