Much of the finance and economics literature studies market equilibria based on the assumption that all market participants have the same beliefs about the economy. This assumption, however, can easily be questioned in reality. Recently, a growing body of literature is examining what happens when market participants have heterogeneous beliefs. This thesis makes two distinct contributions to this area.

The pioneering paper of Harrison and Kreps showed in 1978 how the heterogeneity of investor beliefs can drive speculation, leading the price of an asset to exceed its intrinsic value. By focusing on an extremely simple market model – a finite-state Markov chain – the analysis of Harrison and Kreps achieved great clarity but limited realism. In chapter 2 of this thesis, we achieve similar clarity with greater realism, by considering an asset whose dividend rate is a mean-reverting stochastic process. Our risk-neutral investors agree on the volatility, but have different beliefs about the mean reversion rate. We determine the minimum equilibrium price explicitly; in addition, we characterize it as the unique classical solution of a certain linear differential equation. Our example shows, in a simple and transparent manner, how heterogeneous beliefs about the mean reversion rate can lead to everlasting speculation and a permanent "price bubble."

Recently, a number of papers have examined the equilibria of markets in which the participating investors have heterogeneous beliefs. Virtually all this work relies on Cox-Huang martingale method. Chapter 3 of this thesis develops a different, PDE-based approach. Our analysis is restricted to a very basic model, in which the economy consists of a risk-free bond and a risky dividend-paying asset whose dividend rate is a lognormal stochastic process. There are two investors with CRRA preferences; they agree on the volatility, but have different beliefs about the mean growth rate of the dividend. We show that the equilibrium can be characterized by solving a pair of linear PDEs; moreover, the dynamics of all the relevant economic quantities can be made entirely explicit. Our stochastic-control-based method offers a fresh viewpoint on the study of market equilibria.