金融市场用的数学方法 英文PDF电子书下载

Part Ⅰ Continuous Path Processes 3

1 Continuous-Path Random Processes:Mathematical Prerequisites 3

1.1 Some Definitions 3

1.1.1 Measurability 3

1.1.2 Monotone Class Theorem 4

1.1.3 Probability Measures 5

1.1.4 Filtration 5

1.1.5 Law of a Random Variable,Expectation 6

1.1.6 Independence 6

1.1.7 Equivalent Probabilities and Radon-Nikod？m Densities 7

1.1.8 Construction of Simple Probability Spaces 8

1.1.9 Conditional Expectation 9

1.1.10 Stochastic Processes 10

1.1.11 Convergence 12

1.1.12 Laplace Transform 13

1.1.13 Gaussian Processes 15

1.1.14 Markov Processes 15

1.1.15 Uniform Integrability 18

1.2 Martingales 19

1.2.1 Definition and Main Properties 19

1.2.2 Spaces of Martingales 21

1.2.3 Stopping Times 21

1.2.4 Local Martingales 25

1.3 Continuous Semi-martingales 27

1.3.1 Brackets of Continuous Local Martingales 27

1.3.2 Brackets of Continuous Semi-martingales 29

1.4 Brownian Motion 30

1.4.1 One-dimensional Brownian Motion 30

1.4.2 d-dimensional Brownian Motion 34

1.4.3 Correlated Brownian Motions 34

1.5 Stochastic Calculus 35

1.5.1 Stochastic Integration 36

1.5.2 Integration by Parts 38

1.5.3 It？'s Formula:The Fundamental Formula of Stochastic Calculus 38

1.5.4 Stochastic Differential Equations 43

1.5.5 Stochastic Differential Equations:The Onedimensional Case 47

1.5.6 Partial Differential Equations 51

1.5.7 Doléans-Dade Exponential 52

1.6 Predictable Representation Property 55

1.6.1 Brownian Motion Case 55

1.6.2 Towards a General Definition of the Predictable Representation Property 57

1.6.3 Dudley's Theorem 60

1.6.4 Backward Stochastic Differential Equations 61

1.7 Change of Probability and Girsanov's Theorem 66

1.7.1 Change of Probability 66

1.7.2 Decomposition of P-Martingales as Q-semi-martingales 68

1.7.3 Girsanov's Theorem:The One-dimensional Brownian Motion Case 69

1.7.4 Multidimensional Case 72

1.7.5 Absolute Continuity 73

1.7.6 Condition for Martingale Property of Exponential Local Martingales 74

1.7.7 Predictable Representation Property under a Change of Probability 77

1.7.8 An Example of Invariance of BM under Change of Measure 78

2 Basic Concepts and Examples in Finance 79

2.1 A Semi-martingale Framework 79

2.1.1 The Financial Market 80

2.1.2 Arbitrage Opportunities 83

2.1.3 Equivalent Martingale Measure 85

2.1.4 Admissible Strategies 85

2.1.5 Complete Market 87

2.2 A Diffusion Model 89

2.2.1 Absence of Arbitrage 90

2.2.2 Completeness of the Market 90

2.2.3 PDE Evaluation of Contingent Claims in a Complete Market 92

2.3 The Black and Scholes Model 93

2.3.1 The Model 94

2.3.2 European Call and Put Options 97

2.3.3 The Greeks 101

2.3.4 General Case 102

2.3.5 Dividend Paying Assets 102

2.3.6 R？le of Information 104

2.4 Change of Numéraire 105

2.4.1 Change of Numéraire and Black-Scholes Formula 106

2.4.2 Self-financing Strategy and Change of Numéraire 107

2.4.3 Change of Numéraire and Change of Probability 108

2.4.4 Forward Measure 108

2.4.5 Self-financing Strategies:Constrained Strategies 109

2.5 Feynman-Kac 112

2.5.1 Feynman-Kac Formula 112

2.5.2 Occupation Time for a Brownian Motion 113

2.5.3 Occupation Time for a Drifted Brownian Motion 114

2.5.4 Cumulative Options 116

2.5.5 Quantiles 118

2.6 Ornstein-Uhlenbeck Processes and Related Processes 119

2.6.1 Definition and Properties 119

2.6.2 Zero-coupon Bond 123

2.6.3 Absolute Continuity Relationship for Generalized Vasicek Processes 124

2.6.4 Square of a Generalized Vasicek Process 127

2.6.5 Powers of δ-Dimensional Radial OU Processes,Alias CIR Processes 128

2.7 Valuation of European Options 129

2.7.1 The Garman and Kohlhagen Model for Currency Options 129

2.7.2 Evaluation of an Exchange Option 130

2.7.3 Quanto Options 132

3 Hitting Times:A Mix of Mathematics and Finance 135

3.1 Hitting Times and the Law of the Maximum for Brownian Motion 136

3.1.1 The Law of the Pair of Random Variables(Wt,Mt) 136

3.1.2 Hitting Times Process 138

3.1.3 Law of the Maximum of a Brownian Motion over[0,t] 139

3.1.4 Laws of Hitting Times 140

3.1.5 Law of the Infimum 142

3.1.6 Laplace Transforms of Hitting Times 143

3.2 Hitting Times for a Drifted Brownian Motion 145

3.2.1 Joint Laws of (MX,X) and (mX,X) at Time t 145

3.2.2 Laws of Maximum,Minimum,and Hitting Times 147

3.2.3 Laplace Transforms 148

3.2.4 Computation of W(v)(ll{Ty(X)<t}e-λTy(X)) 149

3.2.5 Normal Inverse Gaussian Law 150

3.3 Hitting Times for Geometric Brownian Motion 151

3.3.1 Laws of the Pairs(MSt,St)and(mSt,St) 151

3.3.2 Laplace Transforms 152

3.3.3 Computation of E(e-λTa(S)ll{Ta(S)<t}) 153

3.4 Hitting Times in Other Cases 153

3.4.1 Ornstein-Uhlenbeck Processes 153

3.4.2 Deterministic Volatility and Nonconstant Barrier 154

3.5 Hitting Time of a Two-sided Barrier for BM and GBM 156

3.5.1 Brownian Case 156

3.5.2 Drifted Brownian Motion 159

3.6 Barrier Options 160

3.6.1 Put-Call Symmetry 160

3.6.2 Binary Options and △'s 163

3.6.3 Barrier Options:General Characteristics 164

3.6.4 Valuation and Hedging of a Regular Down-and-In Call Option When the Underlying is a Martingale 166

3.6.5 Mathematical Results Deduced from the Previous Approach 169

3.6.6 Valuation and Hedging of Regular Down-and-In Call Options:The General Case 172

3.6.7 Valuation and Hedging of Reverse Barrier Options 175

3.6.8 The Emerging Calls Method 177

3.6.9 Closed Form Expressions 178

3.7 Lookback Options 179

3.7.1 Using Binary Options 179

3.7.2 Traditional Approach 180

3.8 Double-barrier Options 182

3.9 Other Options 183

3.9.1 Options Involving a Hitting Time 183

3.9.2 Boost Options 184

3.9.3 Exponential Down Barrier Option 186

3.10 A Structural Approach to Default Risk 188

3.10.1 Merton's Model 188

3.10.2 First Passage Time Models 190

3.11 American Options 191

3.11.1 American Stock Options 192

3.11.2 American Currency Options 193

3.11.3 Perpetual American Currency Options 195

3.12 Real Options 198

3.12.1 Optimal Entry with Stochastic Investment Costs 198

3.12.2 Optimal Entry in the Presence of Competition 201

3.12.3 Optimal Entry and Optimal Exit 204

3.12.4 Optimal Exit and Optimal Entry in the Presence of Competition 205

3.12.5 Optimal Entry and Exit Decisions 206

4 Complements on Brownian Motion 211

4.1 Local Time 211

4.1.1 A Stochastic Fubini Theorem 211

4.1.2 Occupation Time Formula 211

4.1.3 An Approximation of Local Time 213

4.1.4 Local Times for Semi-martingales 214

4.1.5 Tanaka's Formula 214

4.1.6 The Balayage Formula 216

4.1.7 Skorokhod's Reflection Lemma 217

4.1.8 Local Time of a Semi-martingale 222

4.1.9 Generalized It？-Tanaka Formula 226

4.2 Applications 227

4.2.1 Dupire's Formula 227

4.2.2 Stop-Loss Strategy 229

4.2.3 Knock-out BOOST 230

4.2.4 Passport Options 232

4.3 Bridges,Excursions,and Meanders 232

4.3.1 Brownian Motion Zeros 232

4.3.2 Excursions 232

4.3.3 Laws of Tx,dt and gt 233

4.3.4 Laws of(Bt,gt,dt) 236

4.3.5 Brownian Bridge 237

4.3.6 Slow Brownian Filtrations 241

4.3.7 Meanders 242

4.3.8 The Azéma Martingale 243

4.3.9 Drifted Brownian Motion 244

4.4 Parisian Options 246

4.4.1 The Law of(G-,eD(W),WG-,eD) 249

4.4.2 Valuation of a Down-and-In Parisian Option 252

4.4.3 PDE Approach 256

4.4.4 American Parisian Options 257

5 Complements on Continuous Path Processes 259

5.1 Time Changes 259

5.1.1 Inverse of an Increasing Process 259

5.1.2 Time Changes and Stopping Times 260

5.1.3 Brownian Motion and Time Changes 261

5.2 Dual Predictable Projections 264

5.2.1 Definitions 264

5.2.2 Examples 266

5.3 Diffusions 269

5.3.1 (Time-homogeneous)Diffusions 270

5.3.2 Scale Function and Speed Measure 270

5.3.3 Boundary Points 273

5.3.4 Change of Time or Change of Space Variable 275

5.3.5Recurrence 277

5.3.6 Resolvent Kernel and Green Function 277

5.3.7 Examples 279

5.4 Non-homogeneous Diffusions 281

5.4.1 Kolmogorov's Equations 281

5.4.2 Application:Dupire's Formula 284

5.4.3 Fokker-Planck Equation 286

5.4.4 Valuation of Contingent Claims 289

5.5 Local Times for a Diffusion 290

5.5.1 Various Definitions of Local Times 290

5.5.2 Some Diffusions Involving Local Time 291

5.6 Last Passage Times 294

5.6.1 Notation and Basic Results 294

5.6.2 Last Passage Time of a Transient Diffusion 294

5.6.3 Last Passage Time Before Hitting a Level 297

5.6.4 Last Passage Time Before Maturity 298

5.6.5 Absolutely Continuous Compensator 301

5.6.6 Time When the Supremum is Reached 302

5.6.7 Last Passage Times for Particular Martingales 303

5.7 Pitman's Theorem about(2Mt-Wt) 306

5.7.1 Time Reversal of Brownian Motion 306

5.7.2 Pitman's Theorem 307

5.8 Filtrations 309

5.8.1 Strong and Weak Brownian Filtrations 310

5.8.2 Some Examples 312

5.9 Enlargements of Filtrations 315

5.9.1 Immersion of Filtrations 315

5.9.2 The Brownian Bridge as an Example of Initial Enlargement 318

5.9.3 Initial Enlargement:General Results 319

5.9.4 Progressive Enlargement 323

5.10 Filtering the Information 329

5.10.1 Independent Drift 329

5.10.2 Other Examples of Canonical Decomposition 330

5.10.3 Innovation Process 331

6 A Special Family of Diffusions:Bessel Processes 333

6.1 Definitions and First Properties 333

6.1.1 The Euclidean Norm of the n-Dimensional Brownian Motion 333

6.1.2 General Definitions 334

6.1.3 Path Properties 337

6.1.4 Infinitesimal Generator 337

6.1.5 Absolute Continuity 339

6.2 Properties 342

6.2.1 Additivity of BESQ's 342

6.2.2 Transition Densities 343

6.2.3 Hitting Times for Bessel Processes 345

6.2.4 Lamperti's Theorem 347

6.2.5 Laplace Transforms 349

6.2.6 BESQ Processes with Negative Dimensions 353

6.2.7 Squared Radial Ornstein-Uhlenbeck 356

6.3 Cox-Ingersoll-Ross Processes 356

6.3.1 CIR Processes and BESQ 357

6.3.2 Transition Probabilities for a CIR Process 358

6.3.3 CIR Processes as Spot Rate Models 359

6.3.4 Zero-coupon Bond 361

6.3.5 Inhomogeneous CIR Process 364

6.4 Constant Elasticity of Variance Process 365

6.4.1 Particular Case μ=0 366

6.4.2 CEV Processes and CIR Processes 368

6.4.3 CEV Processes and BESQ Processes 368

6.4.4 Properties 370

6.4.5 Scale Functions for CEV Processes 371

6.4.6 Option Pricing in a CEV Model 372

6.5 Some Computations on Bessel Bridges 373

6.5.1 Bessel Bridges 373

6.5.2 Bessel Bridges and Ornstein-Uhlenbeck Processes 374

6.5.3 European Bond Option 376

6.5.4 American Bond Options and the CIR Model 378

6.6 Asian Options 381

6.6.1 Parity and Symmetry Formulae 382

6.6.2 Laws of A(v)θ and A(v)t 383

6.6.3 The Moments of At 388

6.6.4 Laplace Transform Approach 389

6.6.5 PDE Approach 391

6.7 Stochastic Volatility 392

6.7.1 Black and Scholes Implied Volatility 392

6.7.2 A General Stochastic Volatility Model 392

6.7.3 Option Pricing in Presence of Non-normality of Returns:The Martingale Approach 393

6.7.4 Hull and White Model 396

6.7.5 Closed-form Solutions in Some Correlated Cases 398

6.7.6 PDE Approach 401

6.7.7 Heston's Model 401

6.7.8 Mellin Transform 403

Part Ⅱ Jump Processes 407

7 Default Risk:An Enlargement of Filtration Approach 407

7.1 A Toy Model 407

7.1.1 Defaultable Zero-coupon with Payment at Maturity 408

7.1.2 Defaultable Zero-coupon with Payment at Hit 410

7.2 Toy Model and Martingales 412

7.2.1 Key Lemma 412

7.2.2 The Fundamental Martingale 412

7.2.3 Hazard Function 413

7.2.4 Incompleteness of the Toy Model,non Arbitrage Prices 415

7.2.5 Predictable Representation Theorem 415

7.2.6 Risk-neutral Probability Measures 416

7.2.7 Partial Information:Duffie and Lando's Model 418

7.3 Default Times with a Given Stochastic Intensity 418

7.3.1 Construction of Default Time with a Given Stochastic Intensity 418

7.3.2 Conditional Expectation with Respect to Ft 419

7.3.3 Enlargements of Filtrations 420

7.3.4 Conditional Expectations with Respect to ？t 420

7.3.5 Conditional Expectations of F∞-Measurable Random Variables 422

7.3.6 Correlated Defaults:Copula Approach 423

7.3.7 Correlated Defaults:Jarrow and Yu's Model 425

7.4 Conditional Survival Probability Approach 426

7.4.1 Conditional Expectations 427

7.5 Conditional Survival Probability Approach and Immersion 428

7.5.1 (H)-Hypothesis and Arbitrages 429

7.5.2 Pricing Contingent Claims 430

7.5.3 Correlated Defaults:Kusuoka's Example 431

7.5.4 Stochastic Barrier 432

7.5.5 Predictable Representation Theorems 432

7.5.6 Hedging Contingent Claims with DZC 434

7.6 General Case:Without the(H)-Hypothesis 437

7.6.1 An Example of Partial Observation 437

7.6.2 Two Defaults,Trivial Reference Filtration 440

7.6.3 Initial Times 442

7.6.4 Explosive Defaults 444

7.7 Intensity Approach 445

7.7.1 Definition 445

7.7.2 Valuation Formula 446

7.8 Credit Default Swaps 446

7.8.1 Dynamics of the CDS's Price in a single name setting 447

7.8.2 Dynamics of the CDS's Price in a multi-name setting 448

7.9 PDE Approach for Hedging Defaultable Claims 449

7.9.1 Defaultable Asset with Total Default 449

7.9.2 PDE for Valuation 450

7.9.3 General Case 454

8 Poisson Processes and Ruin Theory 457

8.1 Counting Processes and Stochastic Integrals 457

8.2 Standard Poisson Process 459

8.2.1 Definition and First Properties 459

8.2.2 Martingale Properties 461

8.2.3 Infinitesimal Generator 464

8.2.4 Change of Probability Measure:An Example 465

8.2.5 Hitting Times 466

8.3 Inhomogeneous Poisson Processes 467

8.3.1 Definition 467

8.3.2 Martingale Properties 467

8.3.3 Watanabe's Characterization of Inhomogeneous Poisson Processes 468

8.3.4 Stochastic Calculus 469

8.3.5 Predictable Representation Property 473

8.3.6 Multidimensional Poisson Processes 474

8.4 Stochastic Intensity Processes 475

8.4.1 Doubly Stochastic Poisson Processes 475

8.4.2 Inhomogeneous Poisson Processes with Stochastic Intensity 476

8.4.3 It？'s Formula 476

8.4.4 Exponential Martingales 477

8.4.5 Change of Probability Measure 478

8.4.6 An Elementary Model of Prices Involving Jumps 479

8.5 Poisson Bridges 480

8.5.1 Definition of the Poisson Bridge 480

8.5.2 Harness Property 481

8.6 Compound Poisson Processes 483

8.6.1 Definition and Properties 483

8.6.2 Integration Formula 484

8.6.3 Martingales 485

8.6.4 It？'s Formula 492

8.6.5 Hitting Times 492

8.6.6 Change of Probability Measure 494

8.6.7 Price Process 495

8.6.8 Martingale Representation Theorem 496

8.6.9 Option Pricing 497

8.7 Ruin Process 497

8.7.1 Ruin Probability 497

8.7.2 Integral Equation 498

8.7.3 An Example 498

8.8 Marked Point Processes 501

8.8.1 Random Measure 501

8.8.2 Definition 501

8.8.3 An Integration Formula 503

8.8.4 Marked Point Processes with Intensity and Associated Martingales 503

8.8.5 Girsanov's Theorem 504

8.8.6 Predictable Representation Theorem 504

8.9 Poisson Point Processes 505

8.9.1 Poisson Measures 505

8.9.2 Point Processes 506

8.9.3 Poisson Point Processes 506

8.9.4 The It？ Measure of Brownian Excursions 507

9 General Processes:Mathematical Facts 509

9.1 Some Basic Facts about càdlàg Processes 509

9.1.1 An Illustrative Lemma 509

9.1.2 Finite Variation Processes,Pure Jump Processes 510

9.1.3 Some σ-algebras 512

9.2 Stochastic Integration for Square Integrable Martingales 513

9.2.1 Square Integrable Martingales 513

9.2.2 Stochastic Integral 516

9.3 Stochastic Integration for Semi-martingales 517

9.3.1 Local Martingales 517

9.3.2 Quadratic Covariation and Predictable Bracket of Two Local Martingales 519

9.3.3 Orthogonality 521

9.3.4 Semi-martingales 522

9.3.5 Stochastic Integration for Semi-martingales 524

9.3.6 Quadratic Covariation of Two Semi-martingales 525

9.3.7 Particular Cases 525

9.3.8 Predictable Bracket of Two Semi-martingales 527

9.4 It？'s Formula and Girsanov's Theorem 528

9.4.1 It？'s Formula:Optional and Predictable Forms 528

9.4.2 Semi-martingale Local Times 531

9.4.3 Exponential Semi-martingales 532

9.4.4 Change of Probability, Girsanov's Theorem 534

9.5 Existence and Uniqueness of the e.m.m 537

9.5.1 Predictable Representation Property 537

9.5.2 Necessary Conditions for Existence 538

9.5.3 Uniqueness Property 542

9.5.4 Examples 543

9.6 Self-financing Strategies and Integration by Parts 544

9.6.1 The Model 545

9.6.2 Self-financing Strategies and Change of Numéraire 545

9.7 Valuation in an Incomplete Market 547

9.7.1 Replication Criteria 548

9.7.2 Choice of an Equivalent Martingale Measure 549

9.7.3 Indifference Prices 550

10 Mixed Processes 551

10.1 Definition 551

10.2 It？'s Formula 552

10.2.1 Integration by Parts 552

10.2.2 It？'s Formula:One-dimensional Case 553

10.2.3 Multidimensional Case 555

10.2.4 Stochastic Differential Equations 556

10.2.5 Feynman-Kac Formula 557

10.2.6 Predictable Representation Theorem 558

10.3 Change of Probability 559

10.3.1 Exponential Local Martingales 559

10.3.2 Girsanov's Theorem 560

10.4 Mixed Processes in Finance 561

10.4.1 Computation of the Moments 561

10.4.2 Symmetry 562

10.4.3 Hitting Times 563

10.4.4 Affine Jump-Diffusion Model 565

10.4.5 General Jump-Diffusion Processes 569

10.5 Incompleteness 569

10.5.1 The Set of Risk-neutral Probability Measures 570

10.5.2 The Range of Prices for European Call Options 572

10.5.3 General Contingent Claims 575

10.6 Complete Markets with Jumps 578

10.6.1 A Three Assets Model 578

10.6.2 Structure Equations 579

10.7 Valuation of Options 582

10.7.1 The Valuation of European Options 584

10.7.2 American Option 586

11 Lévy Processes 591

11.1 Infinitely Divisible Random Variables 592

11.1.1 Definition 592

11.1.2 Self-decomposable Random Variables 596

11.1.3 Stable Random Variables 598

11.2 Lévy Processes 599

11.2.1 Definition and Main Properties 599

11.2.2 Poisson Point Processes,Lévy Measures 601

11.2.3 Lévy-Khintchine Formula for a Lévy Process 606

11.2.4 It？'s Formulae for a One-dimensional Lévy Process 612

11.2.5 It？'s Formula for Lévy-It？ Processes 613

11.2.6 Martingales 615

11.2.7 Harness Property 620

11.2.8 Representation Theorem of Martingales in a Lévy Setting 621

11.3 Absolutely Continuous Changes of Measures 623

11.3.1 Esscher Transform 623

11.3.2 Preserving the Lévy Property with Absolute Continuity 625

11.3.3 General Case 627

11.4 Fluctuation Theory 628

11.4.1 Maximum and Minimum 628

11.4.2 Pecherskii-Rogozin Identity 631

11.5 Spectrally Negative Lévy Processes 632

11.5.1 Two-sided Exit Times 632

11.5.2 Laplace Exponent of the Ladder Process 633

11.5.3 D.Kendall's Identity 633

11.6 Subordinators 634

11.6.1 Definition and Examples 634

11.6.2 Lévy Characteristics of a Subordinated Process 636

11.7 Exponential Lévy Processes as Stock Price Processes 636

11.7.1 Option Pricing with Esscher Transform 636

11.7.2 A Differential Equation for Option Pricing 637

11.7.3 Put-call Symmetry 638

11.7.4 Arbitrage and Completeness 639

11.8 Variance-Gamma Model 639

11.9 Valuation of Contingent Claims 641

11.9.1 Perpetual American Options 641

A List of Special Features,Probability Laws,and Functions 647

A.1 Main Formulae 647

A.1.1 Absolute Continuity Relationships 647

A.1.2 Bessel Processes 648

A.1.3 Brownian Motion 649

A.1.4 Diffusions 650

A.1.5 Finance 650

A.1.6 Girsanov's Theorem 651

A.1.7 Hitting Times 651

A.1.8 It？'s Formulae 651

A.1.9 Lévy Processes 653

A.1.10 Semi-martingales 654

A.2 Processes 655

A.3 Some Main Models 655

A.4 Some Important Probability Distributions 656

A.4.1 Laws with Density 656

A.4.2 Some Algebraic Properties for Special r.v.'s 656

A.4.3 Poisson Law 657

A.4.4 Gamma and Inverse Gaussian Law 658

A.4.5 Generalized Inverse Gaussian and Normal Inverse Gaussian 659

A.4.6 Variance Gamma VG(σ,ν,θ) 661

A.4.7 Tempered Stable TS(Y±,C±,M±) 661

A.5 Special Functions 662

A.5.1 Gamma and Beta Functions 662

A.5.2 Bessel Functions 662

A.5.3 Hermite Functions 663

A.5.4 Parabolic Cylinder Functions 663

A.5.5 Airy Function 663

A.5.6 Kummer Functions 664

A.5.7 Whittaker Functions 664

A.5.8 Some Laplace Transforms 665

References 667

B Some Papers and Books on Specific Subjects 709

B.1 Theory of Continuous Processes 709

B.1.1 Books 709

B.1.2 Stochastic Differential Equations 709

B.1.3 Backward SDE 709

B.1.4 Martingale Representation Theorems 710

B.1.5 Enlargement of Filtrations 710

B.1.6 Exponential Functionals 710

B.1.7 Uniform Integrability of Martingales 710

B.2 Particular Processes 710

B.2.1 Ornstein-Uhlenbeck Processes 710

B.2.2 CIR Processes 710

B.2.3 CEV Processes 711

B.2.4 Bessel Processes 711

B.3 Processes with Discontinuous Paths 711

B.3.1 Some Books 711

B.3.2 Survey Papers 711

B.4 Hitting Times 711

B.5 Lévy Processes 712

B.5.1 Books 712

B.5.2 Some Papers 712

B.6 Some Books on Finance 712

B.6.1 Discrete Time 712

B.6.2 Continuous Time 712

B.6.3 Collective Books 712

B.6.4 History 713

B.7 Arbitrage 713

B.8 Exotic Options 713

B.8.1 Books 713

B.8.2 Articles 713

Index of Authors 715

Index of Symbols 723

Subject Index 725