arXiv subjects

Torsten Trimborn

Publications and source records attributed to Torsten Trimborn.

12 recordsLinked to original sources

arXiv2021

Mean-field control variate methods for kinetic equations with uncertainties and applications to socio-economic sciences

In this paper, we extend a recently introduced multi-fidelity control variate for the uncertainty quantification of the Boltzmann equation to the case of kinetic models arising in the study of multiagent systems. For these phenomena, where the effect of uncertainties is particularly evident, several models have been developed whose equilibrium states are typically unknown. In particular, we aim to develop efficient numerical methods based on solving the kinetic equations in the phase space by Direct Simulation Monte Carlo (DSMC) coupled to a Monte Carlo sampling in the random space. To this end, exploiting the knowledge of the corresponding mean-field approximation we develop novel mean-field Control Variate (MFCV) methods that are able to strongly reduce the variance of the standard Monte Carlo sampling method in the random space. We verify these observations with several numerical examples based on classical models , including wealth exchanges and opinion formation model for collective phenomena.

math.NA↗

arXiv2020

Simplified ResNet approach for data driven prediction of microstructure-fatigue relationship

The heterogeneous microstructure in metallic components results in locally varying fatigue strength. Metal fatigue strongly depends on size and shape of non-metallic inclusions and pores, commonly referred to as "defects". Nodular cast iron (NCI) contains graphite inclusions (nodules) whose shape and frequency influence the fatigue strength. Fatigue strength can be simulated by micromechanical finite element models. The drawback of these models are the large computational costs. Therefore, we employ a data-driven machine learning methodology. More precisely, we utilize the simplified residual neural network (SimResNet) which was recently introduced (Herty et al., Kinetic Theory for Residual Neural Networks, 2020) to predict fatigue strength from metallographic data. For the training, we use fatigue data which is simulated with a micromechanical model and the shakedown theorem. The micromechanical models are derived directly from micrographs of nodular cast iron, respectively. The application of SimResNet shows a good performance to predict fatigue strength by local microstructures of nodular cast iron. We show several test cases. The simplified character of SimResNet enables fast predictions of fatigue by microstructures, even in comparision to classical residual neural networks.

cs.CE↗

arXiv2020

Novel Insights in the Levy-Levy-Solomon Agent-Based Economic Market Model

The Levy-Levy-Solomon model (A microscopic model of the stock market: cycles, booms, and crashes, Economic Letters 45 (1))is one of the most influential agent-based economic market models. In several publications this model has been discussed and analyzed. Especially Lux and Zschischang (Some new results on the Levy, Levy and Solomon microscopic stock market model, Physica A, 291(1-4)) have shown that the model exhibits finite-size effects. In this study we extend existing work in several directions. First, we show simulations which reveal finite-size effects of the model. Secondly, we shed light on the origin of these finite-size effects. Furthermore, we demonstrate the sensitivity of the Levy-Levy-Solomon model with respect to random numbers. Especially, we can conclude that a low-quality pseudo random number generator has a huge impact on the simulation results. Finally, we study the impact of the stopping criteria in the market clearance mechanism of the Levy-Levy-Solomon model.

q-fin.TR↗

arXiv2019

From Disequilibrium Markets to Equilibrium

The modeling of financial markets as disequilibrium models by ordinary differential equations has become a popular modeling tool. One famous example of such a model is the Beja-Goldman model(The Journal of Finance, 1980) which we consider in this paper. We study the passage from disequilibrium dynamics to equilibrium. Mathematically, this limit corresponds to an asymptotic limit also known as a Tikhonov-Fenichel reduction. Furthermore, we analyze the stability of the reduced equilibrium model and discuss the economic implications. We conduct several numerical examples to visualize and support our analysis.

econ.GN↗

arXiv2019

Stylized Facts and Agent-Based Modeling

The existence of stylized facts in financial data has been documented in many studies. In the past decade the modeling of financial markets by agent-based computational economic market models has become a frequently used modeling approach. The main purpose of these models is to replicate stylized facts and to identify sufficient conditions for their creations. In this paper we introduce the most prominent examples of stylized facts and especially present stylized facts of financial data. Furthermore, we given an introduction to agent-based modeling. Here, we not only provide an overview of this topic but introduce the idea of universal building blocks for agent-based economic market models.

q-fin.GN↗

arXiv2019

Microscopic Derivation of Mean Field Game Models

Mean field game theory studies the behavior of a large number of interacting individuals in a game theoretic setting and has received a lot of attention in the past decade (Lasry and Lions, Japanese journal of mathematics, 2007). In this work, we derive mean field game partial differential equation systems from deterministic microscopic agent dynamics. The dynamics are given by a particular class of ordinary differential equations, for which an optimal strategy can be computed (Bressan, Milan Journal of Mathematics, 2011). We use the concept of Nash equilibria and apply the dynamic programming principle to derive the mean field limit equations and we study the scaling behavior of the system as the number of agents tends to infinity and find several mean field game limits. Especially we avoid in our derivation the notion of measure derivatives. Novel scales are motivated by an example of an agent-based financial market model.

math.OC↗

arXiv2019

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

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.

q-fin.TR↗

arXiv2018

Simulation of Stylized Facts in Agent-Based Computational Economic Market Models

We study the qualitative and quantitative appearance of stylized facts in several agent-based computational economic market (ABCEM) models. We perform our simulations with the SABCEMM (Simulator for Agent-Based Computational Economic Market Models) tool recently introduced by the authors (Trimborn et al. 2019). Furthermore, we present novel ABCEM models created by recombining existing models and study them with respect to stylized facts as well. This can be efficiently performed by the SABCEMM tool thanks to its object-oriented software design. The code is available on GitHub (Trimborn et al. 2018), such that all results can be reproduced by the reader.

econ.GN↗

arXiv2018

A Macroscopic Portfolio Model: From Rational Agents to Bounded Rationality

We introduce a microscopic model of interacting financial agents, where each agent is characterized by two portfolios; money invested in bonds and money invested in stocks. Furthermore, each agent is faced with an optimization problem in order to determine the optimal asset allocation. The stock price evolution is driven by the aggregated investment decision of all agents. In fact, we are faced with a differential game since all agents aim to invest optimal. Mathematically such a problem is ill posed and we introduce the concept of Nash equilibrium solutions to ensure the existence of a solution. Especially, we denote an agent who solves this Nash equilibrium exactly a rational agent. As next step we use model predictive control to approximate the control problem. This enables us to derive a precise mathematical characterization of the degree of rationality of a financial agent. This is a novel concept in portfolio optimization and can be regarded as a general approach. In a second step we consider the case of a fully myopic agent, where we can solve the optimal investment decision of investors analytically. We select the running cost to be the expected missed revenue of an agent and we assume quadratic transaction costs. More precisely the expected revenues are determined by a combination of a fundamentalist or chartist strategy. Then we derive the mean field limit of the microscopic model in order to obtain a macroscopic portfolio model. The novelty in comparison to existent macroeconomic models in literature is that our model is derived from microeconomic dynamics. The resulting portfolio model is a three dimensional ODE system which enables us to derive analytical results. Simulations reveal, that our model is able to replicate the most prominent features of financial markets, namely booms and crashes.

q-fin.PM↗

arXiv2018

SABCEMM-A Simulator for Agent-Based Computational Economic Market Models

We introduce the simulation tool SABCEMM (Simulator for Agent-Based Computational Economic Market Models) for agent-based computational economic market (ABCEM) models. Our simulation tool is implemented in C++ and we can easily run ABCEM models with several million agents. The object-oriented software design enables the isolated implementation of building blocks for ABCEM models, such as agent types and market mechanisms. The user can design and compare ABCEM models in a unified environment by recombining existing building blocks using the XML-based SABCEMM configuration file. We introduce an abstract ABCEM model class which our simulation tool is built upon. Furthermore, we present the software architecture as well as computational aspects of SABCEMM. Here, we focus on the efficiency of SABCEMM with respect to the run time of our simulations. We show the great impact of different random number generators on the run time of ABCEM models. The code and documentation is published on GitHub at https://github.com/SABCEMM/SABCEMM, such that all results can be reproduced by the reader.

q-fin.CP↗

arXiv2017

Portfolio Optimization and Model Predictive Control: A Kinetic Approach

In this paper, we introduce a large system of interacting financial agents in which each agent is faced with the decision of how to allocate his capital between a risky stock or a risk-less bond. The investment decision of investors, derived through an optimization, drives the stock price. The model has been inspired by the econophysical Levy-Levy-Solomon model (Economics Letters, 45). The goal of this work is to gain insights into the stock price and wealth distribution. We especially want to discover the causes for the appearance of power-laws in financial data. We follow a kinetic approach similar to (D. Maldarella, L. Pareschi, Physica A, 391) and derive the mean field limit of our microscopic agent dynamics. The novelty in our approach is that the financial agents apply model predictive control (MPC) to approximate and solve the optimization of their utility function. Interestingly, the MPC approach gives a mathematical connection between the two opponent economic concepts of modeling financial agents to be rational or boundedly rational. Furthermore, this is to our knowledge the first kinetic portfolio model which considers a wealth and stock price distribution simultaneously. Due to our kinetic approach, we can study the wealth and price distribution on a mesoscopic level. The wealth distribution is characterized by a lognormal law. For the stock price distribution, we can either observe a lognormal behavior in the case of long-term investors or a power-law in the case of high-frequency trader. Furthermore, the stock return data exhibits a fat-tail, which is a well known characteristic of real financial data.

q-fin.PM↗

arXiv2017

Mean Field Limit of a Behavioral Financial Market Model

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 model by Cross, Grinfeld, Lamba and Seaman (Physica A, 354) 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.

q-fin.TR↗