Numerical solution of the Hamilton-Jacobi-Bellman formulation for continuous time mean variance asset allocation under stochastic volatility

1 We present efficient partial differential equation (PDE) methods for continuous time mean2 variance portfolio allocation problems when the underlying risky asset follows a stochastic 3 volatility process. The standard formulation for mean variance optimal portfolio allocation 4 problems gives rise to a two-dimensional non-linear Hamilton-Jacobi-Bellman (HJB) PDE. We 5 use a wide stencil method based on a local coordinate rotation (Ma and Forsyth, 2014) to con6 struct a monotone scheme. Furthermore, by using a semi-Lagrangian timestepping method to 7 discretize the drift term and an improved linear interpolation method, accurate efficient frontiers 8 are constructed. This scheme can be shown to be convergent to the viscosity solution of the 9 HJB equation, and the correctness of the proposed numerical framework is verified by numerical 10 examples. We also discuss the effects on the efficient frontier of the stochastic volatility model 11 parameters. 12