Deterministic Income with Deterministic and Stochastic Interest Rates

TL;DR

This paper analyzes optimal consumption strategies for individuals with deterministic income and interest rates modeled as Ornstein-Uhlenbeck processes, providing explicit solutions in certain cases and employing viscosity solutions in others.

Contribution

It introduces explicit formulas for optimal consumption under deterministic income with both deterministic and stochastic interest rates, extending previous models.

Findings

Explicit value function expressions for deterministic and stochastic interest rate models.

Optimal strategies derived for finite and infinite horizons.

Application of viscosity solutions in complex stochastic cases.

Abstract

We consider an individual or household endowed with an initial capital and an income, modeled as a deterministic process with a continuous drift rate. At first, we model the discounting rate as the price of a zero-coupon bond at zero under the assumption of a short rate evolving as an Ornstein-Uhlenbeck process. Then, a geometric Brownian motion as the preference function and an Ornstein-Uhlenbeck process as the short rate are taken into consideration. It is assumed that the primal interest of the economic agent is to maximise the cumulated value of (expected) discounted consumption from a given time up to a finite deterministic time horizon $T\in\R_+$ or, in a stochastic setting, infinite time horizon. We find an explicit expression for the value function and for the optimal strategy in the first two cases. In the third case, we have to apply the viscosity ansatz.