金融数学中的带跳随机微分方程数值解 英文PDF电子书下载

1 Stochastic Differential Equations with Jumps 1

1.1 Stochastic Processes 1

1.2 Supermartingales and Martingales 16

1.3 Quadratic Variation and Covariation 23

1.4 It？Integral 26

1.5 It？Formula 34

1.6 Stochastic Differential Equations 38

1.7 Linear SDEs 45

1.8 SDEs with Jumps 53

1.9 Existence and Uniqueness of Solutions of SDEs 57

1.10 Exercises 59

2 Exact Simulation of Solutions of SDEs 61

2.1 Motivation of Exact Simulation 61

2.2 Sampling from Transition Distributions 63

2.3 Exact Solutions of Multi-dimensional SDEs 78

2.4 Functions of Exact Solutions 99

2.5 Almost Exact Solutions by Conditioning 105

2.6 Almost Exact Simulation by Time Change 113

2.7 Functionals of Solutions of SDEs 123

2.8 Exercises 136

3 Benchmark Approach to Finance and Insurance 139

3.1 Market Model 139

3.2 Best Performing Portfolio 142

3.3 Supermartingale Property and Pricing 145

3.4 Diversification 149

3.5 Real World Pricing Under Some Models 158

3.6 Real World Pricing Under the MMM 168

3.7 Binomial Option Pricing 176

3.8 Exercises 185

4 Stochastic Expansions 187

4.1 Introduction to Wagner-Platen Expansions 187

4.2 Multiple Stochastic Integrals 195

4.3 Coefficient Functions 202

4.4 Wagner-Platen Expansions 206

4.5 Moments of Multiple Stochastic Integrals 211

4.6 Exercises 230

5 Introduction to Scenario Simulation 233

5.1 Approximating Solutions of ODEs 233

5.2 Scenario Simulation 245

5.3 Strong Taylor Schemes 252

5.4 Derivative-Free Strong Schemes 266

5.5 Exercises 271

6 Regular Strong Taylor Approximations with Jumps 273

6.1 Discrete-Time Approximation 273

6.2 Strong Order 10 Taylor Scheme 278

6.3 Commutativity Conditions 286

6.4 Convergence Results 289

6.5 Lemma on Multiple It？Integrals 292

6.6 Proof of the Convergence Theorem 302

6.7 Exercises 307

7 Regular Strong It？Approximations 309

7.1 Explicit Regular Strong Schemes 309

7.2 Drift-Implicit Schemes 316

7.3 Balanced Implicit Methods 321

7.4 Predictor-Corrector Schemes 326

7.5 Convergence Results 331

7.6 Exercises 346

8 Jump-Adapted Strong Approximations 347

8.1 Introduction to Jump-Adapted Approximations 347

8.2 Jump-Adapted Strong Taylor Schemes 350

8.3 Jump-Adapted Derivative-Free Strong Schemes 355

8.4 Jump-Adapted Drift-Implicit Schemes 356

8.5 Predictor-Corrector Strong Schemes 359

8.6 Jump-Adapted Exact Simulation 361

8.7 Convergence Results 362

8.8 Numerical Results on Strong Schemes 368

8.9 Approximation of Pure Jump Processes 375

8.10 Exercises 388

9 Estimating Discretely Observed Diffusions 389

9.1 Maximum Likelihood Estimation 389

9.2 Discretization of Estimators 393

9.3 Transform Functions for Diffusions 397

9.4 Estimation of Affine Diffusions 404

9.5 Asymptotics of Estimating Functions 409

9.6 Estimating Jump Diffusions 413

9.7 Exercises 417

10 Filtering 419

10.1 Kalman-Bucy Filter 419

10.2 Hidden Markov Chain Filters 424

10.3 Filtering a Mean Reverting Process 433

10.4 Balanced Method in Filtering 447

10.5 A Benchmark Approach to Filtering in Finance 456

10.6 Exercises 475

11 Monte Carlo Simulation of SDEs 477

11.1 Introduction to Monte Carlo Simulation 477

11.2 Weak Taylor Schemes 481

11.3 Derivative-Free Weak Approximations 491

11.4 Extrapolation Methods 495

11.5 Implicit and Predictor-Corrector Methods 497

11.6 Exercises 504

12 Regular Weak Taylor Approximations 507

12.1 Weak Taylor Schemes 507

12.2 Commutativity Conditions 514

12.3 Convergence Results 517

12.4 Exercises 522

13 Jump-Adapted Weak Approximations 523

13.1 Jump-Adapted Weak Schemes 523

13.2 Derivative-Free Schemes 529

13.3 Predictor-Corrector Schemes 530

13.4 Some Jump-Adapted Exact Weak Schemes 533

13.5 Convergence of Jump-Adapted Weak Taylor Schemes 534

13.6 Convergence of Jump-Adapted Weak Schemes 543

13.7 Numerical Results on Weak Schemes 548

13.8 Exercises 569

14 Numerical Stability 571

14.1 Asymptotic p-Stability 571

14.2 Stability of Predictor-Corrector Methods 576

14.3 Stability of Some Implicit Methods 583

14.4 Stability of Simplified Schemes 586

14.5 Exercises 590

15 Martingale Representations and Hedge Ratios 591

15.1 General Contingent Claim Pricing 591

15.2 Hedge Ratios for One-dimensional Processes 595

15.3 Explicit Hedge Ratios 601

15.4 Martingale Representation for Non-Smooth Payoffs 606

15.5 Absolutely Continuous Payoff Functions 616

15.6 Maximum of Several Assets 621

15.7 Hedge Ratios for Lookback Options 627

15.8 Exercises 635

16 Variance Reduction Techniques 637

16.1 Various Variance Reduction Methods 637

16.2 Measure Transformation Techniques 645

16.3 Discrete-Time Variance Reduced Estimators 658

16.4 Control Variates 669

16.5 HP Variance Reduction 677

16.6 Exercises 694

17 Trees and Markov Chain Approximations 697

17.1 Numerical Effects of Tree Methods 697

17.2 Efficiency of Simplified Schemes 712

17.3 Higher Order Markov Chain Approximations 720

17.4 Finite Difference Methods 734

17.5 Convergence Theorem for Markov Chains 744

17.6 Exercises 753

18 Solutions for Exercises 755

Acknowledgements 781

Bibliographical Notes 783

References 793

Author Index 835

Index 847