Stochastic Differential Equations

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 xPreface xiSymbols and Abbreviations xiii1 Mathematical Foundations 1: Point-Set Concepts, Set and Measure Functions, Normed Linear Spaces, and Integration 11.1 Set Notation and Operations 11.1.1 Sets and Set Inclusion 11.1.2 Set Algebra 21.2 Single-Valued Functions 41.3 Real and Extended Real Numbers 61.4 Metric Spaces 71.5 Limits of Sequences 81.6 Point-Set Theory 101.7 Continuous Functions 121.8 Operations on Sequences of Sets 131.9 Classes of Subsets of Ω 151.9.1 Topological Space 151.9.2 σ-Algebra of Sets and the Borel σ-Algebra 151.10 Set and Measure Functions 171.10.1 Set Functions 171.10.2 Measure Functions 181.10.3 Outer Measure Functions 191.10.4 Complete Measure Functions 211.10.5 Lebesgue Measure 211.10.6 Measurable Functions 231.10.7 Lebesgue Measurable Functions 261.11 Normed Linear Spaces 271.11.1 Space of Bounded Real-Valued Functions 271.11.2 Space of Bounded Continuous Real-Valued Functions 281.11.3 Some Classical Banach Spaces 291.12 Integration 311.12.1 Integral of a Non-negative Simple Function 321.12.2 Integral of a Non-negative Measurable Function Using Simple Functions 331.12.3 Integral of a Measurable Function 331.12.4 Integral of a Measurable Function on a Measurable Set 341.12.5 Convergence of Sequences of Functions 352 Mathematical Foundations 2: Probability, Random Variables, and Convergence of Random Variables 372.1 Probability Spaces 372.2 Probability Distributions 422.3 The Expectation of a Random Variable 492.3.1 Theoretical Underpinnings 492.3.2 Computational Considerations 502.4 Moments of a Random Variable 522.5 Multiple Random Variables 542.5.1 The Discrete Case 542.5.2 The Continuous Case 592.5.3 Expectations and Moments 632.5.4 The Multivariate Discrete and Continuous Cases 692.6 Convergence of Sequences of Random Variables 722.6.1 Almost Sure Convergence 732.6.2 Convergence in Lp,p>0 732.6.3 Convergence in Probability 752.6.4 Convergence in Distribution 752.6.5 Convergence of Expectations 762.6.6 Convergence of Sequences of Events 782.6.7 Applications of Convergence of Random Variables 792.7 A Couple of Important Inequalities 80Appendix 2.A The Conditional Expectation E(X|Y) 813 Mathematical Foundations 3: Stochastic Processes, Martingales, and Brownian Motion 853.1 Stochastic Processes 853.1.1 Finite-Dimensional Distributions of a Stochastic Process 863.1.2 Selected Characteristics of Stochastic Processes 883.1.3 Filtrations of A 893.2 Martingales 913.2.1 Discrete-Time Martingales 913.2.1.1 Discrete-Time Martingale Convergence 933.2.2 Continuous-Time Martingales 963.2.2.1 Continuous-Time Martingale Convergence 973.2.3 Martingale Inequalities 973.3 Path Regularity of Stochastic Processes 983.4 Symmetric Random Walk 993.5 Brownian Motion 1003.5.1 Standard Brownian Motion 1003.5.2 BM as a Markov Process 1043.5.3 Constructing BM 1063.5.3.1 BM Constructed from N(0, 1) Random Variables 1063.5.3.2 BM as the Limit of Symmetric Random Walks 1083.5.4 White Noise Process 109Appendix 3.A Kolmogorov Existence Theorem: Another Look 109Appendix 3.B Nondifferentiability of BM 1104 Mathematical Foundations 4: Stochastic Integrals, Itô’s Integral, Itô’s Formula, and Martingale Representation 1134.1 Introduction 1134.2 Stochastic Integration: The Itô Integral 1144.3 One-Dimensional Itô Formula 1204.4 Martingale Representation Theorem 1264.5 Multidimensional Itô Formula 127Appendix 4.A Itô’s Formula 129Appendix 4.B Multidimensional Itô Formula 1305 Stochastic Differential Equations 1335.1 Introduction 1335.2 Existence and Uniqueness of Solutions 1345.3 Linear SDEs 1365.3.1 Strong Solutions to Linear SDEs 1375.3.2 Properties of Solutions 1475.3.3 Solutions to SDEs as Markov Processes 1525.4 SDEs and Stability 154Appendix 5.A Solutions of Linear SDEs in Product Form (Evans, 2013; Gard, 1988) 1595.A.1 Linear Homogeneous Variety 1595.A.2 Linear Variety 161Appendix 5.B Integrating Factors and Variation of Parameters 1625.B.1 Integrating Factors 1635.B.2 Variation of Parameters 1646 Stochastic Population Growth Models 1676.1 Introduction 1676.2 A Deterministic Population Growth Model 1686.3 A Stochastic Population Growth Model 1696.4 Deterministic and Stochastic Logistic Growth Models 1706.5 Deterministic and Stochastic Generalized Logistic Growth Models 1746.6 Deterministic and Stochastic Gompertz Growth Models 1776.7 Deterministic and Stochastic Negative Exponential Growth Models 1796.8 Deterministic and Stochastic Linear Growth Models 1816.9 Stochastic Square-Root Growth Model with Mean Reversion 182Appendix 6.A Deterministic and Stochastic Logistic Growth Models with an Allee Effect 184Appendix 6.B Reducible SDEs 1897 Approximation and Estimation of Solutions to Stochastic Differential Equations 1937.1 Introduction 1937.2 Iterative Schemes for Approximating SDEs 1947.2.1 The EM Approximation 1947.2.2 Strong and Weak Convergence of the EM Scheme 1967.2.3 The Milstein (Second-Order) Approximation 1967.3 The Lamperti Transformation 1997.4 Variations on the EM and Milstein Schemes 2037.5 Local Linearization Techniques 2057.5.1 The Ozaki Method 2057.5.2 The Shoji–Ozaki Method 2077.5.3 The Rate of Convergence of the Local Linearization Method 211Appendix 7.A Stochastic Taylor Expansions 212Appendix 7.B The EM and Milstein Discretizations 2177.B.1 The EM Scheme 2177.B.2 The Milstein Scheme 218Appendix 7.C The Lamperti Transformation 2198 Estimation of Parameters of Stochastic Differential Equations 2218.1 Introduction 2218.2 The Transition Probability Density Function Is Known 2228.3 The Transition Probability Density Function Is Unknown 2278.3.1 Parameter Estimation via Approximation Methods 2288.3.1.1 The EM Routine 2288.3.1.2 The Ozaki Routine 2308.3.1.3 The SO Routine 233Appendix 8.A The ML Technique 235Appendix 8.B The Log-Normal Probability Distribution 238Appendix 8.C The Markov Property, Transitional Densities, and the Likelihood Function of the Sample 239Appendix 8.D Change of Variable 241Appendix A: A Review of Some Fundamental Calculus Concepts 245Appendix B: The Lebesgue Integral 259Appendix C: Lebesgue–Stieltjes Integral 261Appendix D: A Brief Review of Ordinary Differential Equations 263References 275Index 279

"An indispensable resource for students and practitioners with limited exposure tomathematics and statistics, Stochastic Differential Equations: An Introduction withApplications in Population Dynamics Modeling is an excellent fit for advanced under-graduates and beginning graduate students, as well as practitioners who need a gentleintroduction to SDEs" Mathematical Reviews, October 2017