Consumption processes and positively homogeneous projection properties

TL;DR

This paper constructs consumption processes with specific projection properties ensuring positivity and zero end balance, including martingale consumption rates, with applications to income and bonus models.

Contribution

It provides a constructive proof for the existence of consumption processes fulfilling a positively homogeneous projection property, expanding the understanding of such processes in stochastic finance.

Findings

Existence of consumption processes with PHPP under various conditions

Construction methods for these processes in finite and infinite settings

Applications demonstrated in income drawdown and bonus theory

Abstract

We constructively prove the existence of time-discrete consumption processes for stochastic money accounts that fulfill a pre-specified positively homogeneous projection property (PHPP) and let the account always be positive and exactly zero at the end. One possible example is consumption rates forming a martingale under the above restrictions. For finite spaces, it is shown that any strictly positive consumption strategy with restrictions as above possesses at least one corresponding PHPP and could be constructed from it. We also consider numeric examples under time-discrete and -continuous account processes, cases with infinite time horizons and applications to income drawdown and bonus theory.