Time-inconsistent consumption-investment problems in incomplete markets under general discount functions

TL;DR

This paper addresses time-inconsistent consumption-investment problems in incomplete markets with random endowments, providing conditions for equilibrium solutions and establishing their uniqueness through a novel equivalence to time-consistent problems.

Contribution

It introduces a verification framework for open-loop equilibrium pairs and proves their uniqueness by linking the problem to a time-consistent formulation.

Findings

Necessary condition for equilibrium pairs

Verification theorem using coupled FBSDEs

Uniqueness of equilibrium solutions

Abstract

In this paper, we study a time-inconsistent consumption-investment problem with random endowments in a possibly incomplete market under general discount functions. We provide a necessary condition and a verification theorem for an open-loop equilibrium consumption-investment pair in terms of a coupled forward-backward stochastic differential equation. Moreover, we prove the uniqueness of the open-loop equilibrium pair by showing that the original time-inconsistent problem is equivalent to an associated time-consistent one.