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.
Highlights
In recent years, there has been a renewed attention in time inconsistency for optimality problems, as well as financial models
Time inconsistency arises in several optimality problems when the optimal strategy selected at some time s is no longer optimal at time t > s
An important illustration of time-inconsistent problem is the mean-variance selection problem, where the time inconsistency is due to the fact that there is a nonlinear function of the expectation of the final wealth in the objective criterion
Summary
There has been a renewed attention in time inconsistency for optimality problems, as well as financial models. An important illustration of time-inconsistent problem is the mean-variance selection problem, where the time inconsistency is due to the fact that there is a nonlinear function of the expectation of the final wealth in the objective criterion Another important problem which produces a time-inconsistent behavior is the investment-consumption problem with non-exponential discounting. This was the case studied by Strotz (1955–1956), where the time inconsistency arises by the fact that the initial point in time enters in a crucial manner the objective criterion. The second approach consists in the formulation of a time-inconsistent decision problem as a non-cooperative game between incarnations of the decision-maker at different instants of time Nash equilibrium of these strategies are considered to define the new concept of solution to the original problem. By capturing the idea of non-commitment, and letting the commitment period being infinitesimally small, he provided a primitive notion of Nash equilibrium strategy
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