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

## Books 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.

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.

Dedication x

Preface xi

Symbols and Abbreviations xiii

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

1.1 Set Notation and Operations 1

1.1.1 Sets and Set Inclusion 1

1.1.2 Set Algebra 2

1.2 Single-Valued Functions 4

1.3 Real and Extended Real Numbers 6

1.4 Metric Spaces 7

1.5 Limits of Sequences 8

1.6 Point-Set Theory 10

1.7 Continuous Functions 12

1.8 Operations on Sequences of Sets 13

1.9 Classes of Subsets of Ω 15

1.9.1 Topological Space 15

1.9.2 σ-Algebra of Sets and the Borel σ-Algebra 15

1.10 Set and Measure Functions 17

1.10.1 Set Functions 17

1.10.2 Measure Functions 18

1.10.3 Outer Measure Functions 19

1.10.4 Complete Measure Functions 21

1.10.5 Lebesgue Measure 21

1.10.6 Measurable Functions 23

1.10.7 Lebesgue Measurable Functions 26

1.11 Normed Linear Spaces 27

1.11.1 Space of Bounded Real-Valued Functions 27

1.11.2 Space of Bounded Continuous Real-Valued Functions 28

1.11.3 Some Classical Banach Spaces 29

1.12 Integration 31

1.12.1 Integral of a Non-negative Simple Function 32

1.12.2 Integral of a Non-negative Measurable Function Using Simple Functions 33

1.12.3 Integral of a Measurable Function 33

1.12.4 Integral of a Measurable Function on a Measurable Set 34

1.12.5 Convergence of Sequences of Functions 35

2 Mathematical Foundations 2: Probability, Random Variables, and Convergence of Random Variables 37

2.1 Probability Spaces 37

2.2 Probability Distributions 42

2.3 The Expectation of a Random Variable 49

2.3.1 Theoretical Underpinnings 49

2.3.2 Computational Considerations 50

2.4 Moments of a Random Variable 52

2.5 Multiple Random Variables 54

2.5.1 The Discrete Case 54

2.5.2 The Continuous Case 59

2.5.3 Expectations and Moments 63

2.5.4 The Multivariate Discrete and Continuous Cases 69

2.6 Convergence of Sequences of Random Variables 72

2.6.1 Almost Sure Convergence 73

2.6.2 Convergence in Lp,p>0 73

2.6.3 Convergence in Probability 75

2.6.4 Convergence in Distribution 75

2.6.5 Convergence of Expectations 76

2.6.6 Convergence of Sequences of Events 78

2.6.7 Applications of Convergence of Random Variables 79

2.7 A Couple of Important Inequalities 80

Appendix 2.A The Conditional Expectation E(X|Y) 81

3 Mathematical Foundations 3: Stochastic Processes, Martingales, and Brownian Motion 85

3.1 Stochastic Processes 85

3.1.1 Finite-Dimensional Distributions of a Stochastic Process 86

3.1.2 Selected Characteristics of Stochastic Processes 88

3.1.3 Filtrations of A 89

3.2 Martingales 91

3.2.1 Discrete-Time Martingales 91

3.2.1.1 Discrete-Time Martingale Convergence 93

3.2.2 Continuous-Time Martingales 96

3.2.2.1 Continuous-Time Martingale Convergence 97

3.2.3 Martingale Inequalities 97

3.3 Path Regularity of Stochastic Processes 98

3.4 Symmetric Random Walk 99

3.5 Brownian Motion 100

3.5.1 Standard Brownian Motion 100

3.5.2 BM as a Markov Process 104

3.5.3 Constructing BM 106

3.5.3.1 BM Constructed from N(0, 1) Random Variables 106

3.5.3.2 BM as the Limit of Symmetric Random Walks 108

3.5.4 White Noise Process 109

Appendix 3.A Kolmogorov Existence Theorem: Another Look 109

Appendix 3.B Nondifferentiability of BM 110

4 Mathematical Foundations 4: Stochastic Integrals, Itô’s Integral, Itô’s Formula, and Martingale Representation 113

4.1 Introduction 113

4.2 Stochastic Integration: The Itô Integral 114

4.3 One-Dimensional Itô Formula 120

4.4 Martingale Representation Theorem 126

4.5 Multidimensional Itô Formula 127

Appendix 4.A Itô’s Formula 129

Appendix 4.B Multidimensional Itô Formula 130

5 Stochastic Differential Equations 133

5.1 Introduction 133

5.2 Existence and Uniqueness of Solutions 134

5.3 Linear SDEs 136

5.3.1 Strong Solutions to Linear SDEs 137

5.3.2 Properties of Solutions 147

5.3.3 Solutions to SDEs as Markov Processes 152

5.4 SDEs and Stability 154

Appendix 5.A Solutions of Linear SDEs in Product Form (Evans, 2013; Gard, 1988) 159

5.A.1 Linear Homogeneous Variety 159

5.A.2 Linear Variety 161

Appendix 5.B Integrating Factors and Variation of Parameters 162

5.B.1 Integrating Factors 163

5.B.2 Variation of Parameters 164

6 Stochastic Population Growth Models 167

6.1 Introduction 167

6.2 A Deterministic Population Growth Model 168

6.3 A Stochastic Population Growth Model 169

6.4 Deterministic and Stochastic Logistic Growth Models 170

6.5 Deterministic and Stochastic Generalized Logistic Growth Models 174

6.6 Deterministic and Stochastic Gompertz Growth Models 177

6.7 Deterministic and Stochastic Negative Exponential Growth Models 179

6.8 Deterministic and Stochastic Linear Growth Models 181

6.9 Stochastic Square-Root Growth Model with Mean Reversion 182

Appendix 6.A Deterministic and Stochastic Logistic Growth Models with an Allee Effect 184

Appendix 6.B Reducible SDEs 189

7 Approximation and Estimation of Solutions to Stochastic Differential Equations 193

7.1 Introduction 193

7.2 Iterative Schemes for Approximating SDEs 194

7.2.1 The EM Approximation 194

7.2.2 Strong and Weak Convergence of the EM Scheme 196

7.2.3 The Milstein (Second-Order) Approximation 196

7.3 The Lamperti Transformation 199

7.4 Variations on the EM and Milstein Schemes 203

7.5 Local Linearization Techniques 205

7.5.1 The Ozaki Method 205

7.5.2 The Shoji–Ozaki Method 207

7.5.3 The Rate of Convergence of the Local Linearization Method 211

Appendix 7.A Stochastic Taylor Expansions 212

Appendix 7.B The EM and Milstein Discretizations 217

7.B.1 The EM Scheme 217

7.B.2 The Milstein Scheme 218

Appendix 7.C The Lamperti Transformation 219

8 Estimation of Parameters of Stochastic Differential Equations 221

8.1 Introduction 221

8.2 The Transition Probability Density Function Is Known 222

8.3 The Transition Probability Density Function Is Unknown 227

8.3.1 Parameter Estimation via Approximation Methods 228

8.3.1.1 The EM Routine 228

8.3.1.2 The Ozaki Routine 230

8.3.1.3 The SO Routine 233

Appendix 8.A The ML Technique 235

Appendix 8.B The Log-Normal Probability Distribution 238

Appendix 8.C The Markov Property, Transitional Densities, and the Likelihood Function of the Sample 239

Appendix 8.D Change of Variable 241

Appendix A: A Review of Some Fundamental Calculus Concepts 245

Appendix B: The Lebesgue Integral 259

Appendix C: Lebesgue–Stieltjes Integral 261

Appendix D: A Brief Review of Ordinary Differential Equations 263

References 275

Index 279

## Books & Journals

### Books #### Introduction to Computation and Modeling for Differential Equations, 2nd Edition #### Probability and Stochastic Processes 