The Continuous Time Exponential Utility Equilibrium Under Different Beliefs

The topic of this paper is the equilibrium with exponential utility function. Equilibrium studies generally have two aspects, one is the existence of equilibrium, the other is when the equilibrium exists what the properties of the market are presented. This paper will discuss the exponential utility function in different beliefs, existence of equilibrium and properties of equilibrium price.We consider the simple case first: the equilibrium model of two consumers. We assume that the two consumers’ income satisfies a stochastic differential equation(SDE), their own beliefs satisfies a SDE, and establish the equilibrium model of two consumers in satisfy the equilibrium condition. Under the conditions of equilibrium, this paper proves that the equilibrium price is a positive martingale.We extend the model to the case of N consumers, establish the equilibrium model of N consumers, obtain the parameter conditions of equilibrium and SDE of equilibrium price, and prove that it is a positive martingale.