Time-Consistent Investment and Consumption Strategies under a General Discount Function

Abstract

In the present paper, we investigate the Merton portfolio management problem in the context of non-exponential discounting, a context that gives rise to time-inconsistency of the decision-maker. We consider equilibrium policies within the class of open-loop controls that are characterized, in our context, by means of a variational method which leads to a stochastic system that consists of a flow of forward-backward stochastic differential equations and an equilibrium condition. An explicit representation of the equilibrium policies is provided for the special cases of power, logarithmic and exponential utility functions.

Weak equilibria for time‐inconsistent control: With applications to investment‐withdrawal decisions

“…In most of the existing literatures, where analytical solutions are attainable, the resulting equilibrium investment proportions are constant (or at least independent from the wealth). See Karatzas and Wang (2000) for time-consistent stopping control problems, Ekeland and Pirvu (2008), Yong (2012), Björk et al (2017), Alia et al (2021), He and Jiang (2021), among others, for time-inconsistent control problems. Combining these observations, we argue that the proposed model in the present paper, where the agent are supposed to make stopping and control decision simultaneously with the presence of time-inconsistency, brings essential differences from various existing models, and generates nontrivial results.…”

“…(2017), Alia et al. (2021), He and Jiang (2021), among others, for time‐inconsistent control problems. Combining these observations, we argue that the proposed model in the present paper, where the agent are supposed to make stopping and control decision simultaneously with the presence of time‐inconsistency, brings essential differences from various existing models, and generates nontrivial results.…”

“…1 See Karatzas and Wang (2000), Ekeland and Pirvu (2008), Yong (2012), Björk et al (2017), Alia et al (2021), and He and Jiang (2021), among others. 2 Here we are not concerned with what the filtration is.…”

Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems

“…This approach was adopted and further developed by Björk-Khapko-Murgoci [3], Djehiche-Huang [4], Yong [33,34], Wei-Yong-Yu [28], Yan-Yong [29], Wang-Yong [23], Hu-Jin-Zhou [13,14], Hu-Huang-Li [12], Alia [1], to mention a few. Time-inconsistent consumption-investment problems under nonexponential discounting were studied by, for example, Ekeland-Pirvu [5], Alia et al [2], and Hamaguchi [8]. The equilibrium concepts investigated in the literature can be roughly divided into two different types, that is, (i) a closed-loop equilibrium strategy and (ii) an open-loop equilibrium control.…”

“…They characterized an open-loop equilibrium control by using a variational method, which is a natural generalization of the stochastic maximum principle of Peng [16] to the time-inconsistent problem. This approach was further developed in [12,34,29,24,25,2,1,8].…”