An example of a stochastic equilibrium with incomplete markets

TL;DR

This paper establishes the existence and uniqueness of stochastic equilibria in incomplete continuous-time financial markets with heterogeneous agents, using novel PDE stability results and fixed point methods.

Contribution

It introduces a new approach based on PDE stability and fixed point theorems to prove equilibrium existence and uniqueness in incomplete markets with general endowments.

Findings

Proves existence and uniqueness of equilibria in incomplete markets.

Develops a numerical method for computing equilibria.

Shows that equilibrium allocations are not necessarily Pareto optimal.

Abstract

We prove existence and uniqueness of stochastic equilibria in a class of incomplete continuous-time financial environments where the market participants are exponential utility maximizers with heterogeneous risk-aversion coefficients and general Markovian random endowments. The incompleteness featured in our setting - the source of which can be thought of as a credit event or a catastrophe - is genuine in the sense that not only the prices, but also the family of replicable claims itself is determined as a part of the equilibrium. Consequently, equilibrium allocations are not necessarily Pareto optimal and the related representative-agent techniques cannot be used. Instead, we follow a novel route based on new stability results for a class of semilinear partial differential equations related to the Hamilton-Jacobi-Bellman equation for the agents' utility-maximization problems. This approach leads to a reformulation of the problem where the Banach fixed point theorem can be used not only to show existence and uniqueness, but also to provide a simple and efficient numerical procedure for its computation.