Limit Theorems for Individual-Based Models in Economics and Finance

TL;DR

This paper develops a general theoretical framework using limit theorems to analyze how local interactions among agents in economics and finance lead to macroscopic market behaviors, inspired by statistical physics.

Contribution

It introduces a broad, unified approach to derive law of large numbers and central limit theorems for complex interacting particle systems in economics and finance.

Findings

Proves a law of large numbers for measure-valued processes.

Establishes a central limit theorem with Gaussian fluctuations.

Provides a mathematical foundation for macroscopic market modeling.

Abstract

There is a widespread recent interest in using ideas from statistical physics to model certain types of problems in economics and finance. The main idea is to derive the macroscopic behavior of the market from the random local interactions between agents. Our purpose is to present a general framework that encompasses a broad range of models, by proving a law of large numbers and a central limit theorem for certain interacting particle systems with very general state spaces. To do this we draw inspiration from some work done in mathematical ecology and mathematical physics. The first result is proved for the system seen as a measure-valued process, while to prove the second one we will need to introduce a chain of embeddings of some abstract Banach and Hilbert spaces of test functions and prove that the fluctuations converge to the solution of a certain generalized Gaussian stochastic differential equation taking values in the dual of one of these spaces.