金融数学中的带跳随机微分方程数值解 英文2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

金融数学中的带跳随机微分方程数值解 英文PDF格式电子书版下载

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1 Stochastic Differential Equations with Jumps1

1.1 Stochastic Processes1

1.2 Supermartingales and Martingales16

1.3 Quadratic Variation and Covariation23

1.4 It？Integral26

1.5 It？Formula34

1.6 Stochastic Differential Equations38

1.7 Linear SDEs45

1.8 SDEs with Jumps53

1.9 Existence and Uniqueness of Solutions of SDEs57

1.10 Exercises59

2 Exact Simulation of Solutions of SDEs61

2.1 Motivation of Exact Simulation61

2.2 Sampling from Transition Distributions63

2.3 Exact Solutions of Multi-dimensional SDEs78

2.4 Functions of Exact Solutions99

2.5 Almost Exact Solutions by Conditioning105

2.6 Almost Exact Simulation by Time Change113

2.7 Functionals of Solutions of SDEs123

2.8 Exercises136

3 Benchmark Approach to Finance and Insurance139

3.1 Market Model139

3.2 Best Performing Portfolio142

3.3 Supermartingale Property and Pricing145

3.4 Diversification149

3.5 Real World Pricing Under Some Models158

3.6 Real World Pricing Under the MMM168

3.7 Binomial Option Pricing176

3.8 Exercises185

4 Stochastic Expansions187

4.1 Introduction to Wagner-Platen Expansions187

4.2 Multiple Stochastic Integrals195

4.3 Coefficient Functions202

4.4 Wagner-Platen Expansions206

4.5 Moments of Multiple Stochastic Integrals211

4.6 Exercises230

5 Introduction to Scenario Simulation233

5.1 Approximating Solutions of ODEs233

5.2 Scenario Simulation245

5.3 Strong Taylor Schemes252

5.4 Derivative-Free Strong Schemes266

5.5 Exercises271

6 Regular Strong Taylor Approximations with Jumps273

6.1 Discrete-Time Approximation273

6.2 Strong Order 10 Taylor Scheme278

6.3 Commutativity Conditions286

6.4 Convergence Results289

6.5 Lemma on Multiple It？Integrals292

6.6 Proof of the Convergence Theorem302

6.7 Exercises307

7 Regular Strong It？Approximations309

7.1 Explicit Regular Strong Schemes309

7.2 Drift-Implicit Schemes316

7.3 Balanced Implicit Methods321

7.4 Predictor-Corrector Schemes326

7.5 Convergence Results331

7.6 Exercises346

8 Jump-Adapted Strong Approximations347

8.1 Introduction to Jump-Adapted Approximations347

8.2 Jump-Adapted Strong Taylor Schemes350

8.3 Jump-Adapted Derivative-Free Strong Schemes355

8.4 Jump-Adapted Drift-Implicit Schemes356

8.5 Predictor-Corrector Strong Schemes359

8.6 Jump-Adapted Exact Simulation361

8.7 Convergence Results362

8.8 Numerical Results on Strong Schemes368

8.9 Approximation of Pure Jump Processes375

8.10 Exercises388

9 Estimating Discretely Observed Diffusions389

9.1 Maximum Likelihood Estimation389

9.2 Discretization of Estimators393

9.3 Transform Functions for Diffusions397

9.4 Estimation of Affine Diffusions404

9.5 Asymptotics of Estimating Functions409

9.6 Estimating Jump Diffusions413

9.7 Exercises417

10 Filtering419

10.1 Kalman-Bucy Filter419

10.2 Hidden Markov Chain Filters424

10.3 Filtering a Mean Reverting Process433

10.4 Balanced Method in Filtering447

10.5 A Benchmark Approach to Filtering in Finance456

10.6 Exercises475

11 Monte Carlo Simulation of SDEs477

11.1 Introduction to Monte Carlo Simulation477

11.2 Weak Taylor Schemes481

11.3 Derivative-Free Weak Approximations491

11.4 Extrapolation Methods495

11.5 Implicit and Predictor-Corrector Methods497

11.6 Exercises504

12 Regular Weak Taylor Approximations507

12.1 Weak Taylor Schemes507

12.2 Commutativity Conditions514

12.3 Convergence Results517

12.4 Exercises522

13 Jump-Adapted Weak Approximations523

13.1 Jump-Adapted Weak Schemes523

13.2 Derivative-Free Schemes529

13.3 Predictor-Corrector Schemes530

13.4 Some Jump-Adapted Exact Weak Schemes533

13.5 Convergence of Jump-Adapted Weak Taylor Schemes534

13.6 Convergence of Jump-Adapted Weak Schemes543

13.7 Numerical Results on Weak Schemes548

13.8 Exercises569

14 Numerical Stability571

14.1 Asymptotic p-Stability571

14.2 Stability of Predictor-Corrector Methods576

14.3 Stability of Some Implicit Methods583

14.4 Stability of Simplified Schemes586

14.5 Exercises590

15 Martingale Representations and Hedge Ratios591

15.1 General Contingent Claim Pricing591

15.2 Hedge Ratios for One-dimensional Processes595

15.3 Explicit Hedge Ratios601

15.4 Martingale Representation for Non-Smooth Payoffs606

15.5 Absolutely Continuous Payoff Functions616

15.6 Maximum of Several Assets621

15.7 Hedge Ratios for Lookback Options627

15.8 Exercises635

16 Variance Reduction Techniques637

16.1 Various Variance Reduction Methods637

16.2 Measure Transformation Techniques645

16.3 Discrete-Time Variance Reduced Estimators658

16.4 Control Variates669

16.5 HP Variance Reduction677

16.6 Exercises694

17 Trees and Markov Chain Approximations697

17.1 Numerical Effects of Tree Methods697

17.2 Efficiency of Simplified Schemes712

17.3 Higher Order Markov Chain Approximations720

17.4 Finite Difference Methods734

17.5 Convergence Theorem for Markov Chains744

17.6 Exercises753

18 Solutions for Exercises755

Acknowledgements781

Bibliographical Notes783

References793

Author Index835

Index847