Law of Large Numbers for a Heterogeneous System of Stochastic Differential Equations with Strong Local Interaction and Economic Applications

Abstract

A model for the activities of a finite number of agents in an economy is presented as the solution to a system of stochastic differential equations driven by general semimartingales and displaying an extended form of strong local interaction. We demonstrate a law of large numbers for the systems of processes as the number of agents goes to infinity under a weak convergence hypothesis on the triangular array of starting values and driving semimartingales which induces the systems of equations. Further, it is shown that the limit can be uniquely characterized by the distributions of the coordinate processes of the solution to an associated infinite-dimensional stochastic differential equation. Finally, an explicit example describing a currency market is discussed.