# Mean field model for knowledge propagation and economic growth

- Responsibility
- Katerina Velcheva
- Publication
- [Stanford, California] : [Stanford University], 2020
- Copyright notice
- ©2020
- Physical description
- 1 online resource

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## Description

### Creators/Contributors

- Author/Creator
- Velcheva, Katerina Krasimirova, author.
- Contributor
- Papanicolaou, George degree supervisor. Thesis advisor
- Ryzhik, Leonid degree committee member. Thesis advisor
- Ying, Lexing degree committee member. Thesis advisor
- Stanford University. Department of Mathematics.

### Contents/Summary

- Summary
- My dissertation research is on the mathematical analysis of systems of partial differential equations that arise in modeling the equilibrium of mean filed games. I study the existence of traveling wave solutions of such systems and the large time behavior of the associated time dependent problem. Mean field models are used in optimal decision making based on stochastic games with a very large population of agents that are statistically the same. The main advantage of such models is that the overall effect of the other agents on a single one can be replaced by an averaged effect. Hence, the dynamics of a single agent can be determined as a solution of an optimal control problem that depends on the overall distribution of the agents. These models are governed by two partial differential equations, a forward Kolmogorov equation that keeps track of the distribution of agents in the model, and a backward Hamilton Jacobi Bellman (HJB) equation for the value function of the optimal stochastic control problem for each agent. An equilibrium solution to this mean field problem is a solution of the coupled system of the two equations. In my study I focus on an a mean field model for knowledge propagation and economic growth in which interaction between agents is represented by collisions and has diffusion added, in order to get a more realistic model for knowledge propagation. The equilibrium solution of such a mean field model is the solution of a Boltzmann equation for the agent distribution coupled to an HJB equation for the optimal strategy for the control problem. I have studied the existence of a BGP solution of the mean field learning model with diffusion in logarithmic variables. Such a solution can be considered as a generalization of traveling wave solutions of a system of Fisher-KPP (Fisher- Kolmogorov, Petrovsky, Piskunov) type of equation and an HJB equation. In Chapter 2 I prove that a traveling wave solution of the mean field learning model with diffusion exists and I have explored it numerically. The method that is used to construct traveling wave solutions is a variation of a well known method for the construction of traveling wave solutions of Fisher-KPP type equations. The numerical results that I have obtained in Chapter 4 support and extend the theory as many features observed numerically can be proved analytically. I have also studied analytically and numerically the forward-backward system, supplemented by an initial condition for the Fisher-KPP type of equation and a terminal condition for the HJB equation. In chapter 3 I have proved existence of solutions for a fixed terminal time T . In Chapter 4 I have tested the stability of the traveling wave numerically, by using perturbed traveling wave profiles for the initial value and I found that as time increases the solution of the time dependent problem tends to the traveling wave profiles

### Bibliographic information

- Publication date
- 2020
- Copyright date
- 2020
- Note
- Submitted to the Department of Mathematics
- Note
- Thesis Ph.D. Stanford University 2020