# Robust Mathematical Formulation and Probabilistic Description of Agent-Based Computational Economic Market Models

## TL;DR

This paper develops continuous-time formulations and probabilistic descriptions of agent-based economic market models, demonstrating stability, computational efficiency, and the potential of kinetic theory for mesoscopic modeling.

## Contribution

It introduces new continuous-time formulations, proves stability conditions, and explores probabilistic approaches using kinetic theory for two prominent agent-based models.

## Key findings

- Time-scaling impacts Levy-Levy-Solomon model behavior.
- Unnecessary constraint for stability in Franke-Westerhoff model.
- Semi-implicit discretization preserves stability at low computational cost.

## Abstract

In science and especially in economics, agent-based modeling has become a widely used modeling approach. These models are often formulated as a large system of difference equations. In this study, we discuss two aspects, numerical modeling and the probabilistic description for two agent-based computational economic market models: the Levy-Levy-Solomon model and the Franke-Westerhoff model. We derive time-continuous formulations of both models, and in particular we discuss the impact of the time-scaling on the model behavior for the Levy-Levy-Solomon model. For the Franke-Westerhoff model, we proof that a constraint required in the original model is not necessary for stability of the time-continuous model. It is shown that a semi-implicit discretization of the time-continuous system preserves this unconditional stability. In addition, this semi-implicit discretization can be computed at cost comparable to the original model. Furthermore, we discuss possible probabilistic descriptions of time continuous agent-based computational economic market models. Especially, we present the potential advantages of kinetic theory in order to derive mesoscopic desciptions of agent-based models. Exemplified, we show two probabilistic descriptions of the Levy-Levy-Solomon and Franke-Westerhoff model.