# MATHEMATICAL MODELS IN FINANCE mod. 2

## Learning outcomes of the course unit

At the end of the course the students should know some specific tools in order to properly investigate current research topics in the frame of kinetic equations for socio-economic sciences, and they should be able to present contents in a clear way and with a mathematically correct language.

Specifically, these are the expected skills:

- Knowledge and understanding: students will know in detail and will autonomously use mathematical tools in the frame of kinetic models for simple market economies; furthermore they will acquire a good level of understanding of the most recent mathematical contents and theories on the course topics, which enables them to read and understand advanced texts and research articles, and to elaborate then original ideas in specific research contexts.

- Applying knowledge and understanding: students will be able to produce rigorous proofs of (even new) mathematical results and to face new problems in the frame of kinetic equations in socio-economic sciences, proposing new models and studying their properties owing also to appropriate numerical algorithms.

- Making judgements: students will be able to build up non trivial logical proofs, and will be capable to recognize correct or rather fallacious steps of the proofs.

- Communication: students will have to present the main topics of the course in a clear and mathematically correct way.

- Lifelong learning skills: the course will help students to form a flexible mentality allowing them to be employed in work environments that require the ability to face ever-new problems.

At the end of the course the students should know some specific tools in order to properly investigate current research topics in the frame of kinetic equations for socio-economic sciences, and they should be able to present contents in a clear way and with a mathematically correct language.

Specifically, these are the expected skills:

- Knowledge and understanding: students will know in detail and will autonomously use mathematical tools in the frame of kinetic models for simple market economies; furthermore they will acquire a good level of understanding of the most recent mathematical contents and theories on the course topics, which enables them to read and understand advanced texts and research articles, and to elaborate then original ideas in specific research contexts.

- Applying knowledge and understanding: students will be able to produce rigorous proofs of (even new) mathematical results and to face new problems in the frame of kinetic equations in socio-economic sciences, proposing new models and studying their properties owing also to appropriate numerical algorithms.

- Making judgements: students will be able to build up non trivial logical proofs, and will be capable to recognize correct or rather fallacious steps of the proofs.

- Communication: students will have to present the main topics of the course in a clear and mathematically correct way.

- Lifelong learning skills: the course will help students to form a flexible mentality allowing them to be employed in work environments that require the ability to face ever-new problems.

## Course contents summary

Introduction to kinetic equations for a simple market economy.

Investigation (from a modelling and an analytical point of view) of several interaction models for wealth exchange:

- basic deterministic model;

- model with random variables;

- model with taxation and redistribution.

Introduction to kinetic equations for a simple market economy.

Investigation (from a modelling and an analytical point of view) of several interaction models for wealth exchange:

- basic deterministic model;

- model with random variables;

- model with taxation and redistribution.

## Course contents

Wealth distribution function and macroscopic fields of an economic model.

Boltzmann-type evolution equation and its major properties.

Investigation of several interaction models for indiviuals exchanging money:

- basic deterministic model;

- model with random variables taking into account possible non-deterministic effects in the market;

- model with taxation and redistribution of the collected wealth.

We will study existence and properties of a steady state for these models, with particular reference to suitable asymptotic regimes ("continuous trading limit").

We will discuss about the possible formation of distributions with Pareto tails, in agreement with experimental data.

Wealth distribution function and macroscopic fields of an economic model.

Boltzmann-type evolution equation and its major properties.

Investigation of several interaction models for indiviuals exchanging money:

- basic deterministic model;

- model with random variables taking into account possible non-deterministic effects in the market;

- model with taxation and redistribution of the collected wealth.

We will study existence and properties of a steady state for these models, with particular reference to suitable asymptotic regimes ("continuous trading limit").

We will discuss about the possible formation of distributions with Pareto tails, in agreement with experimental data.

## Recommended readings

Books or extended reviews:

- B. During, D. Matthes, G. Toscani, "A Boltzmann-type approach to the formation of wealth distribution curves", Riv. Mat. Univ. Parma 1 (2009) 199–261.

- L. Pareschi, G. Toscani, "Interacting multiagent systems. Kinetic equations and Monte Carlo methods", Oxford University Press (2013).

Research papers:

- A. Chakraborti, B.K. Chakrabarti, "Statistical mechanics of money: how saving propensity affects its distributions", Eur. Phys. J. B. 17 (2000), 167-170.

- S. Cordier, L. Pareschi, G. Toscani, "On a kinetic model for a simple market economy", J. Stat. Phys 120 (2005) 253–277.

- D. Matthes, G. Toscani, "On steady distributions of kinetic models of conservative economies", J. Stat. Phys. 130 (2008), 1087-1117.

- M. Bisi, G. Spiga, G. Toscani, "Kinetic models of conservative economies with wealth redistribution", Comm. Math. Sci. 7 (2009) 901–916.

Books or extended reviews:

- B. During, D. Matthes, G. Toscani, "A Boltzmann-type approach to the formation of wealth distribution curves", Riv. Mat. Univ. Parma 1 (2009) 199–261.

- L. Pareschi, G. Toscani, "Interacting multiagent systems. Kinetic equations and Monte Carlo methods", Oxford University Press (2013).

Research papers:

- A. Chakraborti, B.K. Chakrabarti, "Statistical mechanics of money: how saving propensity affects its distributions", Eur. Phys. J. B. 17 (2000), 167-170.

- S. Cordier, L. Pareschi, G. Toscani, "On a kinetic model for a simple market economy", J. Stat. Phys 120 (2005) 253–277.

- D. Matthes, G. Toscani, "On steady distributions of kinetic models of conservative economies", J. Stat. Phys. 130 (2008), 1087-1117.

- M. Bisi, G. Spiga, G. Toscani, "Kinetic models of conservative economies with wealth redistribution", Comm. Math. Sci. 7 (2009) 901–916.

## Teaching methods

Class lectures.

We hope that Covid Emergency will allow face-to-face classes, the guidelines of University in this respect will be followed.

Class lectures

## Assessment methods and criteria

Oral exam, joint with modulus 1 (the exams of the two parts should be done simultaneously).

For modulus 2, a typical question of the oral exam is a detailed discussion of one of the kinetic models investigated during the course.

Oral exam, joint with modulus 1 (the exams of the two parts should be done simultaneously).

For modulus 2, a typical question of the oral exam is a detailed discussion of one of the kinetic models investigated during the course.

## Other informations

The course "Mathematical Models for Finance" is composed by two modula, which have to be simultaneously chosen by the students. The exam of the two parts will give rise to a unique final grade.

The course "Mathematical Models for Finance" is composed by two modula, which have to be simultaneously chosen by the students. The exam of the two parts will give rise to a unique final grade.