Weakly Nonlinear Analysis of the Hamilton-Jacobi-Bellman Equation Arising from Pension Savings Management

Abstract

The main purpose of this paper is to analyze solutions to a fully nonlinear parabolic equation arising from the problem of optimal portfolio construction. We show how the problem of optimal stock to bond proportion in the management of pension fund portfolio can be formulated in terms of the solution to the Hamilton-Jacobi-Bellman equation. We analyze the solution from qualitative as well as quantitative point of view. We construct useful bounds of solution yielding estimates for the optimal value of the stock to bond proportion in the portfolio. Furthermore we construct asymptotic expansions of a solution in terms of a small model parameter. Finally, we perform sensitivity analysis of the optimal solution with respect to various model parameters and compare analytical results of this paper with the corresponding known results arising from time-discrete dynamic stochastic optimization model.

Keywords: Hamilton-Jacobi-Bellman equation, weakly nonlinear analysis, asymptotic expansion, fully nonlinear parabolic equation, stochastic dynamic programming, pension savings accumulation model

JEL Classification: C15, E27, G19, G11, G23