The Economics of Inaction: Stochastic Control Models with Fixed Costs Book

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 ψ and Ψ 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