Mean field limit of a behavioral financial market model

Abstract

In the past decade there has been a growing interest in agent-based econophysical financial market models. The goal of these models is to gain further insights into stylized facts of financial data. We derive the mean field limit of the econophysical Cross model [7] and show that the kinetic limit is a good approximation of the original model. Our kinetic model is able to replicate some of the most prominent stylized facts, namely fat-tails of asset returns, uncorrelated stock price returns and volatility clustering. Interestingly, psychological misperceptions of investors can be accounted to be the origin of the appearance of stylized facts. The mesoscopic model allows us to study the model analytically. We derive steady state solutions and entropy bounds of the deterministic skeleton. These first analytical results already guide us to explanations for the complex dynamics of the model.This has lead to the new field of research called econophysics which can be traced back to the Dow Jones crash (Black Monday) in 1987. Generally speaking, physicists and economists apply physical theories such as kinetic theory, mean field theory or percolation theory to economic issues. One tool of econophysics are so called agent-based financial market models. Many researchers believe that these models help to gain more insights into financial markets *

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

“…Alternatively, one may derive Boltzmann type equations out of agent dynamics as shown in [40]. Examples of kinetic models derived from agent-based models are [1,2,40,46,48].…”

“…Finally, we emphasize that a time continuous dynamical system may be translated to mesoscopic descriptions modeled using partial differential equations (PDEs) [32,40]. This limit process leading from microscopic dynamics to a mesoscopic description is at the heart of kinetic theory which has been successfully applied to several ABCEM models in the past [1,2,38,46,48]. Thus, one may see this work as a first step from ABCEM models, mostly formulated as difference equations, to financial market models in the physical or mathematical literature, modeled as PDEs.…”

Portfolio optimization and model predictive control: A kinetic approach

“…To overcome these problems, it is possible to derive kinetic models based on partial differential equations (PDEs) out of the microscopic particle models, which give us the possibility to study the appearance of stylized facts analytically. There are several examples of such a kinetic approach in the literature [5,6,9,12,13,17,18,20,22,27,28,33,35,46]. We refer also to [42] for a recent survey.…”