Continuous Time Mean-Variance Optimal Portfolio Allocation

5 We present efficient partial differential equation (PDE) methods for continuous time mean6 variance portfolio allocation problems when the underlying risky asset follows a jump-diffusion. 7 The standard formulation of mean-variance optimal portfolio allocation problems, where the 8 total wealth is the underlying stochastic process, gives rise to a one-dimensional (1-D) non-linear 9 Hamilton-JacobiBellman (HJB) partial integro-differential equation (PIDE) with the control 10 present in the integrand of the jump term, and thus is difficult to solve efficiently. In order to 11 preserve the efficient handling of the jump term, we formulate the asset allocation problem as a 12 2-D impulse control problem, one dimension for each asset in the portfolio, namely the bond 13 and the stock. We then develop a numerical scheme based on a semi-Lagrangian timestepping 14 method, which we show to be monotone, consistent, and stable. Hence, assuming a strong 15 comparison property holds, the numerical solution is guaranteed to converge to the unique 16 viscosity solution of the corresponding HJB PIDE. The correctness of the proposed numerical 17 framework is verified by numerical examples. We also discuss the effects on the efficient frontier 18 of realistic financial modeling, such as different borrowing and lending interest rates, transaction 19 costs and constraints on the portfolio, such as maximum limits on borrowing and solvency. 20