On a Non-Standard Stochastic Control Problem

TL;DR

This paper extends the Merton portfolio problem to include non-exponential discounting, leading to time-inconsistent decisions, and characterizes equilibrium policies through an integral equation, especially for CRRA preferences.

Contribution

It introduces a new approach to find equilibrium policies in a non-standard stochastic control setting with non-exponential discounting, building on prior work by Ekeland and Pirvu.

Findings

Existence of solutions to the integral equation for CRRA preferences.

Characterization of equilibrium policies in non-exponential discounting scenarios.

Extension of previous models to more general discounting functions.

Abstract

This paper considers the Merton portfolio management problem. We are concerned with non-exponential discounting of time and this leads to time inconsistencies of the decision maker. Following Ekeland and Pirvu 2006, we introduce the notion of equilibrium policies and we characterize them by an integral equation. The main idea is to come up with the value function in this context. If risk preferences are of CRRA type, the integral equation which characterizes the value function is shown to have a solution which leads to an equilibrium policy. This work is an extension of Ekeland and Pirvu 2006.