Optimal Consumption Problem in a Diffusion Short-Rate Model

TL;DR

This paper investigates the optimal consumption problem over an infinite horizon within a diffusion short-rate model, establishing existence and uniqueness results, and analyzing specific cases where the value function becomes infinite.

Contribution

It provides a general existence and uniqueness theorem for the optimal consumption problem in diffusion short-rate models, including Vasicek and invariant interval models, and examines cases with infinite value functions.

Findings

Existence and uniqueness of optimal consumption strategies are established.

In certain models, the value function is shown to be infinite.

Specific diffusion models like Vasicek are analyzed in detail.

Abstract

We consider a problem of an optimal consumption strategy on the infinite time horizon when the short-rate is a diffusion process. General existence and uniqueness theorem is illustrated by the Vasicek and so-called invariant interval models. We show also that when the short-rate dynamics is given by a Brownian motion or a geometric Brownian motion, then the value function is infinite.