An incomplete equilibrium with a stochastic annuity

TL;DR

This paper establishes the existence of a continuous-time incomplete Radner equilibrium with stochastic income and a traded annuity, characterized by coupled quadratic backward stochastic differential equations in a Brownian setting.

Contribution

It proves the global existence of such an equilibrium with multidimensional, mean-reverting income processes, extending previous models to include stochastic annuities and incomplete markets.

Findings

Existence of equilibrium under Markovian assumptions

Characterization via coupled quadratic backward SDEs

Inclusion of mean-reverting income dynamics

Abstract

We prove the global existence of an incomplete, continuous-time finite-agent Radner equilibrium in which exponential agents optimize their expected utility over both running consumption and terminal wealth. The market consists of a traded annuity, and, along with unspanned income, the market is incomplete. Set in a Brownian framework, the income is driven by a multidimensional diffusion, and, in particular, includes mean-reverting dynamics. The equilibrium is characterized by a system of fully coupled quadratic backward stochastic differential equations, a solution to which is proved to exist under Markovian assumptions.