Utility Maximization with a Stochastic Clock and an Unbounded Random Endowment

TL;DR

This paper develops a mathematical framework using finitely additive measures to solve utility maximization problems involving stochastic clocks and unbounded endowments, covering classical and complex scenarios.

Contribution

It introduces a new approach with finitely additive measures to establish existence and uniqueness in a broad class of utility maximization problems.

Findings

Established existence and uniqueness for utility maximization with stochastic clocks.

Explicit solution for logarithmic utility with Ornstein-Uhlenbeck local time as a stochastic clock.

Unified treatment of classical and advanced consumption optimization problems.

Abstract

We introduce a linear space of finitely additive measures to treat the problem of optimal expected utility from consumption under a stochastic clock and an unbounded random endowment process. In this way we establish existence and uniqueness for a large class of utility maximization problems including the classical ones of terminal wealth or consumption, as well as the problems depending on a random time-horizon or multiple consumption instances. As an example we treat explicitly the problem of maximizing the logarithmic utility of a consumption stream, where the local time of an Ornstein-Uhlenbeck process acts as a stochastic clock.