Time-consistent annuitization and asset allocation under the mean-variance criterion

We study two continuous-time, time-inconsistent problems for an individual who purchases life annuities and invests her wealth in a risky asset under the mean-variance criterion. In the first problem, the buyer may only purchase life annuities at a bounded, continuous rate, while in the second problem, the buyer may purchase any amount of life annuity income at any time, which results in a singular control problem. We find the individual’s time-consistent equilibrium control strategies explicitly for the two life-annuity problems by solving the corresponding extended Hamilton–Jacobi–Bellman systems of equations. We also discuss the effects of parameters on the equilibrium strategies of the two life-annuity problems.