The Economics of Inaction

Preface ix Chapter 1: Introduction 1 Notes 12 Part I: Mathematical Preliminaries 15 Chapter 2: Stochastic Processes, Brownian Motions, and Diffusions 17 2.1. Random Variables and Stochastic Processes 17 2.2. Independence 18 2.3. Wiener Processes and Brownian Motions 19 2.4. Random Walk Approximation of a Brownian Motion 20 2.5. Stopping Times 24 2.6. Strong Markov Property 24 2.7. Diffusions 25 2.8. Discrete Approximation of an Ornstein-Uhlenbeck Process 27 Notes 28 Chapter 3: Stochastic Integrals and Ito's Lemma 30 3.1. The Hamilton-Jacobi-Bellman Equation 31 3.2. Stochastic Integrals 34 3.3. Ito's Lemma 37 3.4. Geometric Brownian Motion 38 3.5. Occupancy Measure and Local Time 41 3.6. Tanaka's Formula 43 3.7. The Kolmogorov Backward Equation 47 3.8. The Kolmogorov Forward Equation 50 Notes 51 Chapter 4: Martingales 53 4.1. Definition and Examples 53 4.2. Martingales Based on Eigenvalues 57 4.3. The Wald Martingale 58 4.4. Sub- and Supermartingales 60 4.5. Optional Stopping Theorem 63 4.6. Optional Stopping Theorem, Extended 67 4.7. Martingale Convergence Theorem 70 Notes 74 Chapter 5: Useful Formulas for Brownian Motions 75 5.1. Stopping Times Defined by Thresholds 78 5.2. Expected Values for Wald Martingales 79 5.3. The Functions psi and PSI 82 5.4. ODEs for Brownian Motions 87 5.5. Solutions for Brownian Motions When r = 0 88 5.6. Solutions for Brownian Motions When r > 0 93 5.7. ODEs for Diffusions 98 5.8. Solutions for Diffusions When r = 0 98 5.9. Solutions for Diffusions When r > 0 102 Notes 106 Part II: Impulse Control Models 107 Chapter 6: Exercising an Option 109 6.1. The Deterministic Problem 110 6.2. The Stochastic Problem: A Direct Approach 116 6.3. Using the Hamilton-Jacobi-Bellman Equation 119 6.4. An Example 125 Notes 128 Chapter 7: Models with Fixed Costs 129 7.1. A Menu Cost Model 130 7.2. Preliminary Results 133 7.3. Optimizing: A Direct Approach 136 7.4. Using the Hamilton-Jacobi-Bellman Equation 140 7.5. Random Opportunities for Costless Adjustment 145 7.6. An Example 146 Notes 152 Chapter 8: Models with Fixed and Variable Costs 153 8.1. An Inventory Model 154 8.2. Preliminary Results 157 8.3. Optimizing: A Direct Approach 160 8.4. Using the Hamilton-Jacobi-Bellman Equation 162 8.5. Long-Run Averages 164 8.6. Examples 166 8.7. Strictly Convex Adjustment Costs 174 Notes 175 Chapter 9: Models with Continuous Control Variables 176 9.1. Housing and Portfolio Choice with No Transaction Cost 178 9.2. The Model with Transaction Costs 182 9.3. Using the Hamilton-Jacobi-Bellman Equation 184 9.4. Extensions 191 Notes 196 Part III: Instantaneous Control Models 197 Chapter 10: Regulated Brownian Motion 199 10.1. One- and Two-Sided Regulators 201 10.2. Discounted Values 205 10.3. The Stationary Distribution 212 10.4. An Inventory Example 218 Notes 224 Chapter 11: Investment: Linear and Convex Adjustment Costs 225 11.1. Investment with Linear Costs 227 11.2. Investment with Convex Adjustment Costs 232 11.3. Some Special Cases 236 11.4. Irreversible Investment 239 11.5. Irreversible Investment with Two Shocks 243 11.6. A Two-Sector Economy 247 Notes 248 Part IV: Aggregation 251 Chapter 12: An Aggregate Model with Fixed Costs 253 12.1. The Economic Environment 256 12.2. An Economy with Monetary Neutrality 259 12.3. An Economy with a Phillips Curve 261 12.4. Optimizing Behavior and the Phillips Curve 265 12.5. Motivating the Loss Function 278 Notes 280 A Continuous Stochastic Processes 283 A.1. Modes of Convergence 283 A.2. Continuous Stochastic Processes 285 A.3. Wiener Measure 287 A.4. Nondifferentiability of Sample Paths 288 Notes 289 B Optional Stopping Theorem 290 B.1. Stopping with a Uniform Bound, T <= N 290 B.2. Stopping with Pr {T < } = 1 292 Notes 294 References 295 Part Index 303