Stochastic Control of Hereditary Systems and Applications / Edition 1

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Overview

This research monograph develops the Hamilton-Jacobi-Bellman (HJB) theory through dynamic programming principle for a class of optimal control problems for stochastic hereditary differential systems. It is driven by a standard Brownian motion and with a bounded memory or an infinite but fading memory.
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Editorial Reviews

From the Publisher

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"A large class of models from physics, chemistry … etc., is described by shastic hereditary differential equations (SHDEs) driven by a standard Brownian motion. … The monograph is addressed to researchers and advanced graduate students with interest in the theory and applications of optimal control for SHDEs. … The monograph provides a systematic and careful exposition of the fundamental results of the control problems for shastic hereditary differential systems and represents an essential source of information for anyone who wants to work in the field." (Constantin Tudor, Mathematical Reviews, Issue 2009 e)

“The theme of this research monograph is a set of equations that represent a class of infinite-dimensional shastic systems. … This monograph can serve as an introduction and/or a research reference for researchers and advanced graduate students with a special interest in theory and applications of optimal control of SHDEs. The monograph is intended to be as self-contained as possible. … Theory developed in this monograph can be extended with additional efforts to hereditary differential equations driven by semimartingales, such as Lévy processes.” (Adriana Horníková, Technometrics, Vol. 52 (2), May, 2010)

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Product Details

Table of Contents

Preface VII

Notation XV

Introduction and Summary 1

A Basic Notation 2

B Stochastic Control Problems and Summary 3

B1 Optimal Classical Control Problem 5

B2 Optimal Stopping Problem 9

B3 Discrete Approximations 10

B4 Option Pricing 19

B5 Hereditary Portfolio Optimization 23

C Organization of Monograph 34

1 Stochastic Hereditary Differential Equations 37

1.1 Probabilistic Preliminaries 40

1.1.1 Gronwall Inequality 40

1.1.2 Stopping Times 42

1.1.3 Regular Conditional Probability 43

1.2 Brownian Motion and Ito Integrals 45

1.2.1 Brownian Motion 45

1.2.2 Ito Integrals 48

1.2.3 Ito's Formula 52

1.2.4 Girsanov Transformation 53

1.3 SHDE with Bounded Memory 54

1.3.1 Memory Maps 54

1.3.2 The Assumptions 57

1.3.3 Strong Solution 58

1.3.4 Weak Solution 62

1.4 SHDE with Unbounded Memory 66

1.4.1 Memory Maps 67

1.5 Markovian Properties 76

1.6 Conclusions and Remarks 78

2 Stochastic Calculus 79

2.1 Preliminary Analysis on Banach Spaces 79

2.1.1 Bounded Linear and Bilinear Fnctionals 80

2.1.2 Frechet Derivatives 81

2.1.3 C[subscript 0]-Semigroups 82

2.1.4 Bounded and Continuous Functionals on Banach Spaces 85

2.2 The Space C 89

2.3 The Space M 99

2.3.1 The Weighting Function [rho] 100

2.3.2 The S-Operator 102

2.4 Ito and Dynkin Formulas 106

2.4.1 {x[subscript s], s [set membership] [t, T]} 106

2.4.2 {(S(s), S[subscript s]), s [greater than or equal] 0} 112

2.5 Martingale Problem 116

2.6 Conclusions and Remarks 124

3 Optimal Classical Control 127

3.1 Problem Formulation 129

3.1.1 The Controlled SHDE 129

3.1.2 Admissible Controls 132

3.1.3 Statement of the Problem 133

3.2 Existence of Optimal ClassicalControl 134

3.2.1 Admissible Relaxed Controls 137

3.2.2 Existence Result 140

3.3 Dynamic Programming Principle 153

3.3.1 Some Probabilistic Results 153

3.3.2 Continuity of the Value Function 158

3.3.3 The DDP 159

3.4 The Infinite-Dimensional HJB Equation 165

3.5 Viscosity Solution 168

3.6 Uniqueness 175

3.7 Verification Theorems 191

3.8 Finite-Dimensional HJB Equation 192

3.8.1 Special Form of HJB Equation 192

3.8.2 Finite Dimensionality of HJB Equation 195

3.8.3 Examples 199

3.9 Conclusions and Remarks 201

4 Optimal Stopping 203

4.1 The Optimal Stopping Problem 204

4.2 Existence of Optimal Stopping 208

4.2.1 The Infinitesimal Generator 208

4.2.2 An Alternate Formulation 210

4.2.3 Existence and Uniqueness 218

4.3 HJB Variational Inequality 224

4.4 Verification Theorem 226

4.5 Viscosity Solution 228

4.6 A Sketch of a Proof of Theorem 4.5.7 234

4.7 Conclusions and Remarks 244

5 Discrete Approximations 245

5.1 Preliminaries 246

5.1.1 Temporal and Spatial Discretizations 247

5.1.2 Some Lemmas 248

5.2 Semidiscretization Scheme 249

5.2.1 First Approximation Step: Piecewise Constant Segments 250

5.2.2 Second Approximation Step: Piecewise Constant Strategies 257

5.2.3 Overall Discretization Error 264

5.3 Markov Chain Approximation 265

5.3.1 Controlled Markov Chains 267

5.3.2 Optimal Control of Markov Chains 270

5.3.3 Embedding the Controlled Markov Chain 272

5.3.4 Convergence of Approximations 274

5.4 Finite Difference Approximation 278

5.4.1 Finite Difference Scheme 280

5.4.2 Discretization of Segment Functions 289

5.4.3 A Computational Algorithm 291

5.5 Conclusions and Remarks 292

6 Option Pricing 293

6.1 Pricing with Hereditary Structure 297

6.1.1 The Financial Market 297

6.1.2 Contingent Claims 302

6.2 Admissible Trading Strategies 304

6.3 Risk-Neutral Martingale Measures 307

6.4 Pricing of Contingent Claims 309

6.4.1 The European Contingent Claims 311

6.4.2 The American Contingent Claims 313

6.5 Infinite-Dimensional Black-Scholes Equation 314

6.5.1 Equation Derivation 314

6.5.2 Vicosity Solution 320

6.6 HJB Variational Inequality 323

6.7 Series Solution 325

6.7.1 Derivations 325

6.7.2 An Example 328

6.7.3 Convergence of the Series 329

6.7.4 The Algorithm 331

6.8 Conclusions and Remarks 331

7 Hereditary Portfolio Optimization 333

7.1 The Hereditary Portfolio Optimization Problem 336

7.1.1 Hereditary Price Structure with Unbounded Memory 337

7.1.2 The Stock Inventory Space 340

7.1.3 Consumption-Trading Strategies 341

7.1.4 Solvency Region 342

7.1.5 Portfolio Dynamics and Admissible Strategies 344

7.1.6 The Problem Statement 345

7.2 The Controlled State Process 346

7.2.1 The Properties of the Stock Prices 346

7.2.2 Dynkin's Formula for the Controlled State Process 351

7.3 The HJBQVI 352

7.3.1 The Dynamic Programming Principle 352

7.3.2 Derivation of the HJBQVI 353

7.3.3 Boundary Values of the HJBQVI 357

7.4 The Verification Theorem 366

7.5 Properties of Value Function 369

7.5.1 Some Simple Properties 369

7.5.2 Upper Bounds of Value Function 371

7.6 The Viscosity Solution 379

7.7 Uniqueness 390

7.8 Conclusions and Remarks 391

References 393

Index 401

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