Perov's Contraction Principle and Dynamic Programming with Stochastic Discounting

TL;DR

This paper extends contraction principles to vector-valued metrics for dynamic programming with stochastic discounting, enabling broader analysis in asset pricing and savings models.

Contribution

It generalizes Perov's contraction principle and Blackwell's condition to stochastic discount factors using spectral radius conditions.

Findings

Generalized contraction conditions for stochastic discounting

Applied to asset pricing models

Extended Blackwell's sufficient condition

Abstract

This paper shows the usefulness of Perov's contraction principle, which generalizes Banach's contraction principle to a vector-valued metric, for studying dynamic programming problems in which the discount factor can be stochastic. The discounting condition $\beta<1$ is replaced by $\rho(B)<1$, where $B$ is an appropriate nonnegative matrix and $\rho$ denotes the spectral radius. Blackwell's sufficient condition is also generalized in this setting. Applications to asset pricing and optimal savings are discussed.