# Time-consistent investment and consumption strategies under a general discount function

## TL;DR

This paper investigates time-consistent investment and consumption strategies under general discount functions using equilibrium policies, addressing time-inconsistency in non-exponential discounting within a stochastic control framework.

## Contribution

It extends the concept of equilibrium policies to the Merton problem with non-exponential discounting, providing explicit solutions for common utility functions.

## Key findings

- Characterized equilibrium policies via a stochastic system of forward-backward equations.
- Derived explicit equilibrium strategies for power, logarithmic, and exponential utilities.
- Addressed time-inconsistency issues in stochastic portfolio optimization.

## Abstract

The paper [12] examines a concept of equilibrium policies instead of optimal controls in stochastic optimization to analyze a mean-variance portfolio selection problem. We follow the same approach in order to investigate the Merton portfolio management problem in the context of non-exponential discounting, a context that give rise to time-inconsistency of the decision maker. Equilibrium policies are characterized in this 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.