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