[Paper Review] Interacting Agent Feedback Finance Model

Biao Wu|ArXiv.org|Mar 28, 2007

Complex Systems and Time Series Analysis20 references3 citations

TL;DR

This paper proposes an Interacting Agent Feedback Finance Model (IAFFM) that integrates agent-based dynamics with recursive price mechanisms, modeling feedback between agent type distributions and asset prices. It establishes the asymptotic convergence of the empirical distribution of agent types and equilibrium price to a deterministic diffusion process as the number of agents tends to infinity, under weak convergence and continuity assumptions on the model components.

ABSTRACT

We consider a financial market model which consists of a financial asset and a large number of interacting agents classified into many types. Different types of agents are heterogeneous in their price expectations. Each agent can change its type based on the current empirical distribution of the types and the equilibrium price, and the equilibrium price follows a recursive price mechanism based on the previous price and the current empirical distribution of the types. The interaction among the agents, and the interaction between the agents and the equilibrium price, feedback, are modeled. We analyze the asymptotic behavior of the empirical distribution of the types and the equilibrium price when the number of agents goes to infinity. We give a case study of a simple example, and also investigate the fixed points of empirical distribution and equilibrium price of the example.

Motivation & Objective

• To develop a systematic agent-based financial model incorporating feedback between agent behavior and asset prices.
• To analyze the asymptotic behavior of agent type distributions and equilibrium prices in a large-population limit.
• To establish weak convergence of the stochastic process governing agent types and prices to a deterministic diffusion process.
• To investigate fixed points and stability in a simplified example model.
• To provide a rigorous mathematical framework for feedback mechanisms in financial markets using interacting Markov chains and recursive price dynamics.

Proposed method

• Models a financial market with N fixed agents, each assigned to one of r internal states (types), with time evolving in discrete steps.
• Employs a recursive log-price mechanism: ˜qN(k/N) = ˜qN((k−1)/N) + (1/N)gN(k/N, nN(k)/N, ˜qN((k−1)/N)), linking price evolution to current type distribution.
• Introduces agent type transitions via a stochastic matrix P^N, dependent on current type distribution, price, and external environment.
• Analyzes the system in the limit as N → ∞, using weak convergence techniques to show convergence of the empirical measure and price process to a diffusion process.
• Applies martingale problem theory and Kolmogorov’s criterion to prove sample path continuity of the limiting process.
• Uses uniform continuity and boundedness assumptions on gN, ϕN, ψN to establish tightness and convergence of finite-dimensional distributions.

Experimental results

Research questions

• RQ1 How does feedback from the equilibrium price influence the evolution of agent types in a large-agent financial market?
• RQ2 What is the limiting behavior of the empirical distribution of agent types and the asset price as the number of agents tends to infinity?
• RQ3 Under what conditions does the system converge weakly to a deterministic diffusion process?
• RQ4 What are the fixed points of the empirical distribution and equilibrium price in the simplified example model?
• RQ5 How do the interactions between agent types and the price mechanism affect market stability and emergent dynamics?

Key findings

• The empirical distribution of agent types and the equilibrium price process converge weakly to a deterministic diffusion process as the number of agents N → ∞.
• The limiting process satisfies a stochastic differential equation driven by the drift and diffusion coefficients derived from the model's transition rates and price mechanism.
• The limiting process is continuous almost surely, proven via Kolmogorov’s criterion applied to the component processes.
• For the simplified example, fixed points of the empirical distribution and equilibrium price are identified, and their stability is analyzed.
• The convergence is established under uniform boundedness and continuity assumptions on the functions gN, ϕN, and ψN, ensuring tightness and weak convergence.
• The proof relies on compact containment and uniform continuity, with the limiting process satisfying a martingale problem on the space K × R.