Continuous-Time Security Pricing: A Utility Gradient Approach

This paper develops an infinite horizon, continuous-time, single-agent security pricing model. No restrictions are placed on the underlying information filtration (other than the “usual conditions”) and the agent is assumed to have a monotone, concave utility over consumption processes that has a gradient satisfying technical conditions. Equilibrium prices, which are general semimartingales, are shown to be given by the classical formula of total future discounted dividends. The discount factors are given by the gradient’s Riesz representation and are explicitly computed for the case of infinite horizon stochastic differential utilities, additive utilities being a special case. Furthermore, equilibrium price-processes are shown to be unique within a broad class of processes. The method of proof utilizes the connection between the first order Kuhn-Tucker conditions and the martingale property of nominal gain processes.