Existence of stochastic equilibrium with incomplete financial markets

Abstract

This paper analyzes the aritrage-free security markets and the general equilibrium existence problem for a stochastic economy with incomplete financial markets. Information structure is given by an event tree. This paper restricts attention to purely financial securities. It is assume that trading takes place in the sequence of spot markets and futures markets for securities payable in units of account. Unlimited short-selling in securities is allowed. Financial markets may be incomplete: some consumption streams may be impossible to obtain by any trading strategy. Securities may be individually precluded from trade at arbitrary states and dates. The security price process is arbitrage-free the dividend process if and only if there exists a stochstic state price (present value) process: the present value of the security prices at every vertex is the present value of their dividend and capital values over the set of immediate successors; the current value of each security at every vertex is the present value of its future dividend stream over all succeeding vertices. The existence of such an equilibrium is proved under the following condition: continuous, weakly convex, strictly monotone and complete preferences, strictly positive endowments and dividends processes.