Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling

A beginner s guide to stochastic growth modeling

The chief advantage of stochastic growth models over deterministic models is that they combine both deterministic and stochastic elements of dynamic behaviors, such as weather, natural disasters, market fluctuations, and epidemics. This makes stochastic modeling a powerful tool in the hands of practitioners in fields for which population growth is a critical determinant of outcomes.  

However, the background requirements for studying SDEs can be daunting for those who lack the rigorous course of study received by math majors. Designed to be accessible to readers who have had only a few courses in calculus and statistics, this book offers a comprehensive review of the mathematical essentials needed to understand and apply stochastic growth models. In addition, the book describes deterministic and stochastic applications of population growth models including logistic, generalized logistic, Gompertz, negative exponential, and linear.

Ideal for students and professionals in an array of fields including economics, population studies, environmental sciences, epidemiology, engineering, finance, and the biological sciences,  Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling:

Provides precise definitions of many important terms and concepts and provides many solved example problems Highlights the interpretation of results and does not rely on a theorem–proof approach Features comprehensive chapters addressing any background deficiencies readers may have and offers a comprehensive review for those who need a mathematics refresher Emphasizes solution techniques for SDEs and their practical application to the development of stochastic population models

An indispensable resource for students and practitioners with  limited exposure to mathematics and statistics, Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling is an excellent fit for advanced undergraduates and beginning graduate students, as well as practitioners who need a gentle introduction to SDEs.

Dedication

Preface

Chapter 1: Mathematical Foundations 1: Point–Set Concepts, Set and Measure Functions, Normed Linear Spaces, and Integration

1.1 Set Notation and Operations

1.2 Single–Valued Functions

1.3 Real and Extended Real Numbers

1.4 Metric Spaces

1.5 Limits of Sequences

1.6 Point–Set Theory

1.7 Continuous Functions

1.8 Operations on Sequences of Sets

1.9 Classes of Subsets of 1.10 Set and Measure Functions

1.11 Normed Linear Spaces

1.12 Integration

Chapter 2: Mathematical Foundations 2: Probability, Random Variables, and Convergence of Random Variables

2.1 Probability Spaces

2.2 Probability Distributions

2.3 The Expectation of a Random Variable

2.4 Moments of a Random Variable

2.5 Multiple Random Variables

2.6 Convergence of Sequences of Random Variables

2.7 A Couple of Important Inequalities

Appendix 2.A The Conditional Expectation E(X|Y)

Chapter 3: Mathematical Foundations 3: Stochastic Processes, Martingales, and Brownian Motion

3.1 Stochastic Processes

3.2 Martingales

3.3 Path Regularity of Stochastic Processes

3.4 Symmetric Random Walk

3.5 Brownian Motion (BM)

Appendix 3.A Kolmogorov Existence Theorem: Another Look

Chapter 4: Mathematical Foundations 4: Stochastic Integrals, Itô s Integral, Itô s Formula, and Martingale Representation

4.1 Introduction

4.2 Stochastic Integration

4.3 One–Dimensional Itô Formula

4.4 Martingale Representation Theorem

4.5 Multi–Dimensional Itô Formula

Appendix 4.A Itô s Formula

Appendix 4.B Multi–Dimensional Itô Formula

Chapter 5: Stochastic Differential Equations (SDEs)

5.1 Introduction

5.2 Existence and Uniqueness of Solutions

5.3 Linear Stochastic Differential Equations

5.4 Stochastic Differential Equations and Stability

Appendix 5.A Solutions of Linear SDEs in Product Form

Appendix 5.B Integrating Factors and Variation of Parameters

Chapter 6: Stochastic Population Growth Models

6.1 Introduction

6.2 A Deterministic Population Growth Model

6.3 A Stochastic Population Growth Model

6.4 Deterministic and Stochastic Logistic Growth Models

6.5 Deterministic and Stochastic Generalized Logistic Growth Models

6.6 Deterministic and Stochastic Gompertz Growth Models

6.7 Deterministic and Stochastic Negative Exponential Growth Models

6.8 Deterministic and Stochastic Linear Growth Models

6.9 Stochastic Square–Root Growth Models with Mean Reversion

Appendix 6.A Deterministic and Stochastic Logistic Growth Models with an Allee Effect

Appendix 6.B Reducible Stochastic Differential Equations

Chapter 7 : Approximation and Estimation of Solutions to SDEs

7.1 Introduction

7.2 Iterative Schemes for Approximating SDEs

7.3 The Lamperti Transformation

7.4 Variations on the E–M and Milstein Schemes

7.5 Local Linearization Techniques

Appendix 7.A Stochastic Taylor Expansions

Appendix 7.B The Euler– Maruyama (E–M) and Milstein Discretizations

Appendix 7.C The Lamperti Transformation

Chapter 8: Estimation of Parameters of SDEs

8.1 Introduction

8.2 The Transition Probability Density Function is Known

8.3 The Transition Probability Density Function is Unknown

Appendix 8.A The Maximum Likelihood (ML) Technique

Appendix 8.B The Log–Normal Probability Distribution

Appendix 8.C The Markov Property, Transition Densities, and the Likelihood Function of the Sample

Appendix 8.D Change of Variable

Appendix A A Review of Some Fundamental Calculus Concepts

Appendix B The Lebesgue Integral

Appendix C Lebesgue–Stieltjes (L–S) Integral

Appendix D A Brief Review of Ordinary Differential Equations (ODEs)

References

Symbols and Abbreviations

Index

Michael J. Panik, PhD, is Professor in the Department of Economics, Barney School of Business and Public Administration at the University of Hartford in Connecticut. He received his PhD in Economics from Boston College and is a member of the American Mathematical Society, The American Statistical Association, and The Econometric Society.