Numerical methods for Hamilton Jacobi Bellman equations in finance.

Project level: PhD, Masters

Many popular problems in mathematical finance can be posed in terms of a stochastic optimal control problem, which can then be formulated as nonlinear Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs), or partial integro-differential equations (PIDEs), when the underlying random process is diffusion or jump-diffusion, respectively. There are several challenges in solving these HJB PDEs/PIDEs. First, in general, especially in realistic situations where the controls are constrained, there are no analytical solutions to the HJB PDEs/PIDEs. Second, since the PDEs are nonlinear, the solutions are not necessarily unique. Moreover, these nonlinear PDEs do not usually possess smooth classical, i.e. differentiable, solutions. As a result, we must seek the financially relevant solution of the HJB PDEs, which, in this case, is the viscosity solution of these equations.There are a number of available projects that focus on developing efficient numerical methods which guarantees convergence to the viscosity solution for multi-dimensional HJB PDEs arising in finance. Typical applications include multi-asset passport options with uncertain volatility/correlations, and mean-variance portfolio optimization with stochastic interest rates.