Stochastic Optimization in Continuous Time

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
List of Figures......Page 13
Preface......Page 15
1.2.1 Information Sets and σ-Algebras......Page 21
1.2.2 The Cantor Set......Page 24
1.2.3 Borel–Cantelli Lemmas......Page 25
1.2.4 Distribution Functions and Stochastic Processes......Page 28
1.3.1 Conditional Probability......Page 32
1.3.2 Conditional Expectation......Page 35
1.3.3 Change of Variables......Page 40
1.4 Notes and Further Readings......Page 43
2.2.1 From Random Walks to Wiener Process......Page 44
2.2.2 Some Basic Properties of the Wiener Process......Page 47
2.3.1 Introduction......Page 49
2.3.2 Transition Probability......Page 51
2.3.3 Diffusion Processes......Page 55
2.4 Wiener Processes......Page 58
2.4.1 How to Generate More Wiener Processes......Page 59
2.4.2 Differentiability of Sample Functions......Page 62
2.4.3 Stopping Times......Page 65
2.4.4 The Zero Set......Page 70
2.5 Notes and Further Readings......Page 74
3.1 Introduction......Page 76
3.2.1 Is Sigma (s, Xs) dWs Riemann Integrable?......Page 77
3.2.2 The Choice of Ti Matters......Page 80
3.2.3 In Search of the Class of Functions for Sigma (s, ω)......Page 83
3.3 The Ito Integral......Page 85
3.3.1 Definition......Page 86
3.3.2 Martingales......Page 93
3.4.1 Ito’s Lemma......Page 97
3.4.2 Geometric Brownian Motion......Page 100
3.4.3 Population Dynamics......Page 105
3.4.4 Additive Shocks or Multiplicative Shocks......Page 107
3.4.5 Multiple Sources of Uncertainty......Page 110
3.4.6 Multivariate Ito’s Lemma......Page 113
Euler’s Homogenous Differential Equation......Page 117
On the Heat Equation......Page 118
3.5.2 Black–Scholes Formula......Page 120
The Value of Investment Opportunity......Page 122
The Value an Investment Project......Page 124
3.5.4 Budget Equation for an Investor......Page 125
Real-Valued Functions......Page 127
Vector-Valued Functions......Page 130
3.6 Notes and Further Readings......Page 131
4.1 Introduction......Page 133
An Optimal Growth Problem......Page 134
Derivation of the Bellman Equation......Page 136
General Formulation......Page 139
Dynkin’s Formula......Page 140
The Solution to (4.16) is the Expected Utility......Page 142
Verification Theorem......Page 143
4.2.3 Finite-Horizon Problems......Page 145
Admissible Controls......Page 148
One-Dimensional Controlled Processes......Page 150
Multidimensional Controlled Processes......Page 151
An Infinite-Horizon Model......Page 153
Wealth Effect......Page 155
Mutual Fund Theorem......Page 156
A Three-Asset Model......Page 157
Optimal Portfolio Rules......Page 159
Insurance Premium......Page 160
Dasgupta and Heal (1974)......Page 161
Stochastic Extension......Page 163
The Model in Pindyck (1980)......Page 164
User Cost......Page 167
Dynamic Model without Adjustment Costs......Page 168
Adjustment Costs......Page 169
The Model in Pindyck (1982)......Page 171
Survival Function......Page 172
Is More Information Better? An Anecdote......Page 174
Life Insurance......Page 176
4.4 Extension: Recursive Utility......Page 177
4.4.1 Bellman Equation with Recursive Utility......Page 180
Costate Equation......Page 182
Euler Equation......Page 183
Phase Diagram......Page 184
4.5 Notes and Further Readings......Page 187
5.1 Introduction......Page 189
5.2.1 The Meaning of Each Parameter......Page 190
5.2.2 Closed-Form Representations......Page 191
DARA and IRRA (a > 0, b > 0)......Page 192
CARA (a = 0, b > 0) – Exponential Utility Functions......Page 193
DARA and DRRA (a > 0, b < 0)......Page 194
5.3.1 Linear–Quadratic Models......Page 195
5.3.2 Linear–HARA models......Page 197
CARA Utility Functions......Page 200
CRRA Utility Functions......Page 201
Competitive Firm......Page 202
How to Guess the Solution......Page 204
Implications......Page 205
An Optimal Growth Model......Page 206
How to Guess the Solution......Page 207
5.4 Symmetry......Page 208
5.4.1 Linear–Quadratic Model Revisited......Page 210
CRRA Utility Functions......Page 211
CARA Utility Functions......Page 213
5.4.3 Fischer’s Index Bond Model......Page 215
5.4.4 Life Insurance......Page 216
5.5 The Substitution Method......Page 218
5.6 Martingale Representation Method......Page 220
5.6.1 Girsanov Transformation......Page 221
5.6.2 Example: A Portfolio Problem......Page 223
5.6.3 Which Theta to Choose?......Page 224
5.6.4 A Transformed Problem......Page 226
5.7 Inverse Optimum Method......Page 228
5.7.1 The Inverse Optimal Problem: Certainty Case......Page 229
5.7.2 The Inverse Optimal Problem: Stochastic Case......Page 232
Existence Theorem......Page 233
Classical Formulation......Page 239
General Formulation......Page 240
An Explicit Example......Page 241
5.8 Notes and Further Readings......Page 243
6.1 Introduction......Page 245
6.2.1 Issues and Problems......Page 246
Standard Approach......Page 247
6.2.2 Comparison Theorems......Page 249
Ordinary Differential Equation......Page 250
Stochastic Differential Equation......Page 251
Extending (6.3) to the Whole Real Line......Page 253
Existence and Uniqueness Theorem......Page 255
6.2.4 Inaccessible Boundaries......Page 257
General Setting......Page 258
6.3.2 Viscosity Solutions......Page 263
Square-Root Rule......Page 266
Optimal Scheduling......Page 267
Optimal Scheduling......Page 269
Formulation of Money Demand......Page 271
Properties of Money Demand......Page 272
6.4.3 The Tree-Cutting Problem......Page 275
Without Rotation......Page 276
With Rotation......Page 278
6.5 The Expected Discount Factor......Page 280
Wald Martingales......Page 281
Optional Stopping Theorem......Page 282
6.5.2 One Absorbing Barrier......Page 283
Expected Discounted Cost Up to a Stopping Time......Page 284
Two Absorbing Barriers 0 and b, with…......Page 287
Absorbing Barrier......Page 290
Formulation Demand-for-Money Function......Page 291
Properties of the Money Demand......Page 295
Without Rotation......Page 299
With Rotation......Page 300
6.6.3 Investment Timing......Page 303
6.7 Notes and Further Readings......Page 306
1. FARMLAND INVESTMENT......Page 308
2. FUTURES PRICING......Page 309
3. HABIT FORMATION AND PORTFOLIO SELECTION......Page 311
4. MONEY AND GROWTH......Page 312
5. A STOCHASTICALLY GROWING MONETARY MODEL......Page 315
6. GROWTH AND TRADE......Page 317
7. INTEGRABILITY PROBLEM OF ASSET PRICES......Page 319
8. ENTRY AND EXIT......Page 321
9. INVESTMENT LAGS......Page 322
10. BOND PRICE DYNAMICS......Page 323
11. ROTATION PROBLEMS AND GEOMETRIC GROWTH......Page 325
12. ENDOGENOUS SEEDLING COST......Page 326
Bibliography......Page 329
Index......Page 337