Applied Stochastic Control of Jump Diffusions (Universitext)

The main purpose of the book is to give a rigorous introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusions and their applications. Both the dynamic programming method and the stochastic maximum principle method are discussed, as well as the relation between them. Corresponding verification theorems involving the Hamilton–Jacobi–Bellman equation and/or (quasi-)variational inequalities are formulated. The text emphasises applications, mostly to finance. All the main results are illustrated by examples and exercises appear at the end of each chapter with complete solutions. This will help the reader understand the theory and see how to apply it. The book assumes some basic knowledge of stochastic analysis, measure theory and partial differential equations. The 3rd edition is an expanded and updated version of the 2nd edition, containing recent developments within stochastic control and its applications. Specifically, there is a new chapter devoted to a comprehensive presentation of financial markets modelled by jump diffusions, and one on backward stochastic differential equations and convex risk measures. Moreover, the authors have expanded the optimal stopping and the stochastic control chapters to include optimal control of mean-field systems and stochastic differential games. Preface to the Third Edition 7 Preface to the Second Edition 8 Preface to the First Edition 9 Contents 11 1 Stochastic Calculus with Lévy Processes 15 1.1 Basic Definitions and Results on Lévy Processes 15 1.2 The Itô Formula and Related Results 21 1.3 Lévy Stochastic Differential Equations 25 1.4 The Girsanov Theorem and Applications 28 1.5 Exercises 37 2 Financial Markets Modeled by Jump Diffusions 41 2.1 Market Definitions and Arbitrage 41 2.2 Hedging and Completeness 47 2.3 Option Pricing 54 2.3.1 European Options 54 2.3.2 American Options 57 2.4 Exercises 60 3 Optimal Stopping of Jump Diffusions 68 3.1 A General Formulation and a Verification Theorem 68 3.2 Applications and Examples 72 3.3 Optimal Stopping with Delayed Information 78 3.4 Exercises 84 4 Backward Stochastic Differential Equations and Risk Measures 87 4.1 Examples 87 4.2 General BSDEs with Jumps 90 4.3 Linear BSDEs 92 4.4 Comparison Theorems 94 4.5 Convex Risk Measures and BSDEs 97 4.5.1 Dynamic Risk Measures 99 4.5.2 A Dual Representation of Convex Risk Measures 100 4.6 Exercises 101 5 Stochastic Control of Jump Diffusions 104 5.1 The Dynamic Programming Approach 104 5.2 Stochastic Maximum Principles for Partial Information Control 112 5.2.1 A Sufficient Maximum Principle 114 5.2.2 A Necessary Maximum Principle 116 5.2.3 The Relation Between the Maximum Principle and Dynamic Programming 119 5.2.4 Utility Maximization 120 5.2.5 Mean-Variance Portfolio Selection 122 5.3 The Maximum Principle with Infinite Horizon 126 5.3.1 A Sufficient Maximum Principle 127 5.3.2 A Necessary Maximum Principle 132 5.4 Optimal Control of FBSDEs by Means of Stochastic HJB Equations 135 5.4.1 Optimal Control of FBSDEs 136 5.4.2 Applications in Mathematical Finance 140 5.5 Optimal Control of Stochastic Delay Equations 145 5.5.1 A Sufficient Maximum Principle 148 5.5.2 A Necessary Maximum Principle 151 5.5.3 Time-Advanced BSDEs with Jumps 155 5.5.4 Example: Optimal Consumption from a Cash Flow with Delay 159 5.6 Exercises 161 6 Stochastic Differential Games 167 6.1 Stochastic Differential (Markov) Games, HJB–Isaacs Equations 167 6.1.1 Entropic Risk Minimization Example 170 6.2 Stochastic Maximum Principles 174 6.2.1 General (Non-zero) Stochastic Differential Games 175 6.2.2 The Zero-Sum Game Case 180 6.2.3 Proofs of the Stochastic Maximum Principles 184 6.2.4 Risk Minimization by FBSDE Games 189 6.3 Mean-Field Games 192 6.3.1 Two Motivating Examples 192 6.3.2 General Mean-Field Non-zero Sum Games 194 6.3.3 A Sufficient Maximum Principle 195 6.3.4 A Necessary Maximum Principle 199 6.3.5 Application to Model Uncertainty Control 202 6.3.6 The Zero-Sum Game Case 211 6.3.7 The Single Player Case 214 6.4 Exercises 216 7 Combined Optimal Stopping and Stochastic Control of Jump Diffusions 221 7.1 Introduction 221 7.2 A General Mathematical Formulation 222 7.3 Applications 227 7.4 Exercises 231 8 Singular Control for Jump Diffusions 234 8.1 An Illustrating Example 234 8.2 A General Formulation 236 8.3 Application to Portfolio Optimization with Transaction Costs 242 8.4 Exercises 244 9 Impulse Control of Jump Diffusions 247 9.1 A General Formulation and a Verification Theorem 247 9.2 Examples 252 9.3 Exercises 260 10 Approximating Impulse Control by Iterated Optimal Stopping 263 10.1 Iterative Scheme 263 10.2 Examples 274 10.3 Exercises 279 11 Combined Stochastic Control and Impulse Control of Jump Diffusions 281 11.1 A Verification Theorem 281 11.2 Examples 284 11.3 Iterative Methods 289 11.4 Exercises 290 12 Viscosity Solutions 292 12.1 Viscosity Solutions of Variational Inequalities 293 12.1.1 Uniqueness 296 12.2 The Value Function is Not Always calC1 297 12.3 Viscosity Solutions of HJBQVI 300 12.4 Numerical Analysis of HJBQVI 311 12.4.1 Finite Difference Approximation 311 12.4.2 A Policy Iteration Algorithm for HJBQVI 314 12.5 Exercises 318 13 Optimal Control of Stochastic Partial Differential Equations and Partial (Noisy) Observation Control 320 13.1 A Motivating Example 320 13.2 The Maximum Principle 321 13.2.1 Return to Example13.1 328 13.3 A Necessary Maximum Principle 333 13.4 Controls Which do not Depend on x 336 13.5 Application to Partial (Noisy) Observation Optimal Control 339 13.5.1 Optimal Portfolio with Noisy Observations 343 13.6 Exercises 346 14 Solutions of Selected Exercises 349 14.1 Exercises of Chap. 1 349 14.2 Exercises of Chap. 2 355 14.3 Exercises of Chap. 3 358 14.4 Exercises of Chap. 4 373 14.5 Exercises of Chap. 5 374 14.6 Exercises of Chap. 6 382 14.7 Exercises of Chap. 7 386 14.8 Exercises of Chap. 8 388 14.9 Exercises of Chap. 9 393 14.10 Exercises of Chap. 10 406 14.11 Exercises of Chap. 11 410 14.12 Exercises of Chap. 12 414 14.13 Exercises of Chap. 13 420 References 422 Notation and Symbols 433 Index 436 The main purpose of the book is to give a rigorous introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusions and their applications. Both the dynamic programming method and the stochastic maximum principle method are discussed, as well as the relation between them. Corresponding verification theorems involving the Hamilton-Jacobi-Bellman equation and/or (quasi- )variational inequalities are formulated. The text emphasises applications, mostly to finance. All the main results are illustrated by examples and exercises appear at the end of each chapter with complete solutions. This will help the reader understand the theory and see how to apply it. The book assumes some basic knowledge of stochastic analysis, measure theory and partial differential equations. The 3rd edition is an expanded and updated version of the 2nd edition, containing recent developments within stochastic control and its applications. Specifically, there is a new chapter devoted to a comprehensive presentation of financial markets modelled by jump diffusions, and one on backward stochastic differential equations and convex risk measures. Moreover, the authors have expanded the optimal stopping and the stochastic control chapters to include optimal control of mean-field systems and stochastic differential games The main purpose of the book is to give a rigorous, yet mostly nontechnical, introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusions and its applications. The types of control problems covered include classical stochastic control, optimal stopping, impulse control and singular control. Both the dynamic programming method and the maximum principle method are discussed, as well as the relation between them. Corresponding verification theorems involving the Hamilton-Jacobi Bellman equation and/or (quasi- )variational inequalities are formulated. There are also chapters on the viscosity solution formulation and numerical methods. The text emphasises applications, mostly to finance. All the main results are illustrated by examples and exercises appear at the end of each chapter with complete solutions. This will help the reader understand the theory and see how to apply it. The book assumes some basic knowledge of stochastic analysis, measure theory and partial differential equations. In the 2nd edition there is a new chapter on optimal control of stochastic partial differential equations driven by Lévy processes. There is also a new section on optimal stopping with delayed information. Moreover, corrections and other improvements have been made The main purpose of the book is to give a rigorous, yet mostly nontechnical, introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusions (i.e. solutions of stochastic differential equations driven by Lévy processes) and its applications. The types of control problems covered include classical stochastic control, optimal stopping, impulse control and singular control. Both the dynamic programming method and the maximum principle method are discussed, as well as the relation between them. Corresponding verification theorems involving the Hamilton-Jacobi Bellman equation and/or (quasi-)variational inequalities are formulated. There are also chapters on the viscosity solution formulation and numerical methods. The text emphasises applications, mostly to finance. All the main results are illustrated by examples and exercises appear at the end of each chapter with complete solutions. This will help the reader understand the theory and see how to apply it. The book assumes some basic knowledge of stochastic analysis, measure theory and partial differential equations. "The main purpose of the book is to give a rigorous, yet mostly nontechnical, introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusions and its applications. The types of control problems covered include classical stochastic control, optimal stopping, impulse control and singular control. Both the dynamic programming method and the maximum principle method are discussed, as well as the relation between them. Corresponding verification theorems involving the Hamilton-Jacobi-Bellman equation and/or (quasi- )variational inequalities are formulated. There is also a chapter on the viscosity solution formulation and numerical methods. The text emphasises applications, mostly to finance. All the main results are illustrated by examples and exercises appear at the end of each chapter with complete solutions. This will help the reader understand the theory and see how to apply it. The book assumes some basic knowledge of stochastic analysis, measure theory and partial differential equations."--Jacket Here is a rigorous introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusions and its applications. Discussion includes the dynamic programming method and the maximum principle method, and their relationship. The text emphasises real-world applications, primarily in finance. Results are illustrated by examples, with end-of-chapter exercises including complete solutions. The 2nd edition adds a chapter on optimal control of stochastic partial differential equations driven by Lévy processes, and a new section on optimal stopping with delayed information. Basic knowledge of stochastic analysis, measure theory and partial differential equations is assumed.