Markowitz's mean-variance defined contribution pension fund management under inflation: A continuous-time model

In defined contribution (DC) pension schemes, the financial risk borne by the member occurs during the accumulation phase. To build up sufficient funds for retirement, scheme members invest their wealth in a portfolio of assets. This paper considers an optimal investment problem of a scheme member facing stochastic inflation under the Markowitz mean-variance criterion. Besides, we consider a more general market with multiple assets that can all be risky. By applying the Lagrange method and stochastic dynamic programming techniques, we derive the associated Hamilton-Jacobi-Bellman (HJB) equation, which can be converted into six correlated but relatively simple partial differential equations (PDEs). The explicit solutions for these six PDEs are derived by using the homogenization approach and the variable transformation technique. Then the closed-form expressions for the optimal strategy and the efficient frontier can be obtained through the Lagrange dual theory. In addition, we illustrate the results by some numerical examples.