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**Around the life cycle: deterministic consumption-investment strategies.**
*(English)*
Zbl 1416.91345

Summary: We study a classical continuous-time consumption-investment problem of a power utility investor with deterministic labor income with the important feature that the consumption-investment process is constrained to be deterministic. This is motivated by the design of modern pension schemes of defined contribution type where, typically, the savings rate is constant and the proportional investment in growth stocks is a function of age or time-to-retirement, a so-called life-cycle investment strategy. We derive and study the optimal behavior corresponding to the optimal product design within this realistic family of products with deterministic decision profiles. We also propose a couple of suboptimal deterministic strategies inspired from the optimal stochastic strategy and compare the optimal stochastic control, the optimal deterministic control, and these suboptimal deterministic controls. The conclusion is that only little is lost by constraining to deterministic strategies and only little is lost by implementing the suboptimal simple explicit strategies rather than the optimal one we derive.

### MSC:

91G10 | Portfolio theory |

93E20 | Optimal stochastic control |

49N90 | Applications of optimal control and differential games |

### Keywords:

continuous-time consumption-investment problem; life-cycle investment; optimal stochastic control; optimal deterministic control; suboptimal deterministic controls
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\textit{M. C. Christiansen} and \textit{M. Steffensen}, N. Am. Actuar. J. 22, No. 3, 491--507 (2018; Zbl 1416.91345)

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