A Variational Approach for Mean-Variance-Optimal Deterministic Consumption and Investment

Abstract

A significant number of life insurance contracts are based on deterministic investment strategies-this justifies to restrict the set of admissible controls to deterministic controls. Optimal deterministic controls can be identified by Hamilton-Jacobi-Bellman techniques, but for the corresponding partial differential equations only numerical solutions are available and so the general existence of optimal controls is unclear. We present a non-constructive existence result and derive necessary characterizations for optimal controls by using a Pontryagin maximum principle. Furthermore, based on the variational idea of the Pontryagin maximum principle, we derive a numerical optimization algorithm for the calculation of optimal controls.

Worst-Case Scenario Portfolio Optimization Given the Probability of a Crash

“…Remark 3 This Lemma is important because often an optimization problem is solved only on the set of deterministic strategies (see e.g., Korn and Wilmott [9], Korn and Menkens [10], or Christiansen [6]) and not on the set of stochastic strategies (which include the deterministic ones). This is done because it is often simpler to solve the optimization problem on the set of deterministic strategies.…”

Deterministic Investment Strategy in a Dc Pension Plan With Inflation Risk Under Mean-Variance Criterion

“…They point out that, by optimizing over the deterministic investment and consumption strategy, the troubles of diffusive consumption, diffusive investment proportions, satisfaction points and time-consistency problems can be avoided. By adopting the Pontryagin maximum principle approach, Christiansen [8] present a non-constructive existence result on the deterministic investment and consumption strategies with the meanvariance criterion and derive necessary characterizations for optimal strategies. Under the criterion of power utility maximization, Christiansen and Steffensen [10] investigate a deterministic investment and consumption problem with labor income.…”

“…In reality, the contribution rate is related to the income of the DC plan member. The income processes considered in Christiansen and Steffensen [9,10] and Christiansen [8] are assumed to be deterministic. However, since the income of the DC plan member is often linked to the financial market, it is reasonable to consider a stochastic contribution rate.…”