Stochastic integration theory

019853969X......Page 1
Contents......Page 8
Preface......Page 14
1.1 Random functions......Page 22
1.1.1 Trajectories of stochastic processes......Page 23
1.1.2 Jumps of stochastic processes......Page 24
1.1.3 When are stochastic processes equal?......Page 27
1.2 Measurability of Stochastic Processes......Page 28
1.2.1 Filtration, adapted, and progressively measurable processes......Page 29
1.2.2 Stopping times......Page 34
1.2.3 Stopped variables, σ-algebras, and truncated processes......Page 40
1.2.4 Predictable processes......Page 44
1.3 Martingales......Page 50
1.3.1 Doob’s inequalities......Page 51
1.3.2 The energy equality......Page 56
1.3.3 The quadratic variation of discrete time martingales......Page 58
1.3.4 The downcrossings inequality......Page 63
1.3.5 Regularization of martingales......Page 67
1.3.6 The Optional Sampling Theorem......Page 70
1.3.7 Application: elementary properties of Lévy processes......Page 79
1.3.8 Application: the first passage times of the Wiener processes......Page 101
1.3.9 Some remarks on the usual assumptions......Page 112
1.4 Localization......Page 113
1.4.1 Stability under truncation......Page 114
1.4.2 Local martingales......Page 115
1.4.3 Convergence of local martingales: uniform convergence on compacts in probability......Page 125
1.4.4 Locally bounded processes......Page 127
2 Stochastic Integration with Locally Square-Integrable Martingales......Page 129
2.1 The Itô–Stieltjes Integrals......Page 130
2.1.1 Itô–Stieltjes integrals when the integrators have finite variation......Page 132
2.1.2 Itô–Stieltjes integrals when the integrators are locally square-integrable martingales......Page 138
2.1.3 Itô–Stieltjes integrals when the integrators are semimartingales......Page 145
2.1.5 The integral process......Page 147
2.1.6 Integration by parts and the existence of the quadratic variation......Page 149
2.1.7 The Kunita–Watanabe inequality......Page 155
2.2 The Quadratic Variation of Continuous Local Martingales......Page 159
2.3 Integration when Integrators are Continuous Semimartingales......Page 167
2.3.1 The space of square-integrable continuous local martingales......Page 168
2.3.2 Integration with respect to continuous local martingales......Page 172
2.3.4 The Dominated Convergence Theorem for stochastic integrals......Page 183
2.3.5 Stochastic integration and the Itô–Stieltjes integral......Page 185
2.4.1 The quadratic variation of locally square-integrable martingales......Page 188
2.4.2 Integration when the integrators are locally square-integrable martingales......Page 192
2.4.3 Stochastic integration when the integrators are semimartingales......Page 197
3 The Structure of Local Martingales......Page 200
3.1.1 Predictable stopping times......Page 203
3.1.2 Decomposition of thin sets......Page 209
3.1.3 The extended conditional expectation......Page 211
3.1.4 Definition of the predictable projection......Page 213
3.1.5 The uniqueness of the predictable projection, the predictable section theorem......Page 215
3.1.6 Properties of the predictable projection......Page 222
3.1.7 Predictable projection of local martingales......Page 225
3.1.8 Existence of the predictable projection......Page 227
3.2.1 Predictable Radon–Nikodym Theorem......Page 228
3.2.2 Predictable Compensator of locally integrable processes......Page 234
3.2.3 Properties of the Predictable Compensator......Page 238
3.3 The Fundamental Theorem of Local Martingales......Page 240
3.4 Quadratic Variation......Page 243
4.1 Purely Discontinuous Local Martingales......Page 246
4.1.1 Orthogonality of local martingales......Page 248
4.1.2 Decomposition of local martingales......Page 253
4.1.3 Decomposition of semimartingales......Page 255
4.2 Purely Discontinuous Local Martingales and Compensated Jumps......Page 256
4.2.1 Construction of purely discontinuous local martingales......Page 261
4.2.2 Quadratic variation of purely discontinuous local martingales......Page 265
4.3 Stochastic Integration With Respect To Local Martingales......Page 267
4.3.1 Definition of stochastic integration......Page 269
4.3.2 Properties of stochastic integration......Page 271
4.4 Stochastic Integration With Respect To Semimartingales......Page 275
4.4.1 Integration with respect to special semimartingales......Page 278
4.4.2 Linearity of the stochastic integral......Page 282
4.4.3 The associativity rule......Page 283
4.4.4 Change of measure......Page 285
4.5 The Proof of Davis’ Inequality......Page 298
4.5.1 Discrete-time Davis’ inequality......Page 300
4.5.2 Burkholder’s inequality......Page 308
5.1.1 The proof of the theorem......Page 313
5.1.2 Dellacherie’s formulas and the natural processes......Page 320
5.1.3 The sub- super- and the quasi-martingales are semimartingales......Page 324
5.2 Semimartingales as Good Integrators......Page 329
5.3 Integration of Adapted Product Measurable Processes......Page 335
5.4 Theorem of Fubini for Stochastic Integrals......Page 340
5.5 Martingale Representation......Page 349
6 Itô’s Formula......Page 372
6.1 Itô’s Formula for Continuous Semimartingales......Page 374
6.2.1 Zeros of Wiener processes......Page 380
6.2.2 Continuous Lévy processes......Page 387
6.2.3 Lévy’s characterization of Wiener processes......Page 389
6.2.4 Integral representation theorems for Wiener processes......Page 394
6.2.5 Bessel processes......Page 396
6.3.1 Locally absolutely continuous change of measure......Page 398
6.3.2 Semimartingales and change of measure......Page 399
6.3.3 Change of measure for continuous semimartingales......Page 401
6.3.4 Girsanov’s formula for Wiener processes......Page 403
6.3.5 Kazamaki–Novikov criteria......Page 407
6.4 Itô’s Formula for Non-Continuous Semimartingales......Page 415
6.4.1 Itô’s formula for processes with finite variation......Page 419
6.4.2 The proof of Itô’s formula......Page 422
6.4.3 Exponential semimartingales......Page 432
6.5 Itô’s Formula For Convex Functions......Page 438
6.5.1 Derivative of convex functions......Page 439
6.5.2 Definition of local times......Page 443
6.5.3 Meyer–Itô formula......Page 450
6.5.4 Local times of continuous semimartingales......Page 459
6.5.5 Local time of Wiener processes......Page 466
6.5.6 Ray–Knight theorem......Page 471
6.5.7 Theorem of Dvoretzky Erdos and Kakutani......Page 478
7.1 Lévy processes......Page 481
7.1.1 Poisson processes......Page 482
7.1.2 Compound Poisson processes generated by the jumps......Page 485
7.1.3 Spectral measure of Lévy processes......Page 493
7.1.4 Decomposition of Lévy processes......Page 501
7.1.5 Lévy–Khintchine formula for Lévy processes......Page 507
7.1.6 Construction of Lévy processes......Page 510
7.1.7 Uniqueness of the representation......Page 512
7.2 Predictable Compensators of Random Measures......Page 517
7.2.1 Measurable random measures......Page 518
7.2.2 Existence of predictable compensator......Page 522
7.3 Characteristics of Semimartingales......Page 529
7.4.1 Examples: probability of jumps of processes with independent increments......Page 534
7.4.2 Predictable cumulants......Page 539
7.4.3 Semimartingales with independent increments......Page 544
7.4.4 Characteristics of semimartingales with independent increments......Page 551
7.4.5 The proof of the formula......Page 555
7.5 Decomposition of Processes with Independent Increments......Page 559
A.1 The Monotone Class Theorem......Page 568
A.2 Projection and the Measurable Selection Theorems......Page 571
A.3 Cramér’s Theorem......Page 572
A.4 Interpretation of Stopped σ-algebras......Page 576
B.1 Basic Properties......Page 580
B.2 Existence of Wiener Processes......Page 588
B.3 Quadratic Variation of Wiener Processes......Page 592
C: Poisson processes......Page 600
Notes and Comments......Page 615
References......Page 618
G......Page 624
L......Page 625
P......Page 626
S......Page 627
W......Page 628
Y......Page 629