Stochastic integration theory/ Peter Medvegyev

Contents
1 Stochastic processes 1
1.1 Random functions 1
1.1.1 Trajectories of stochastic processes 2
1.1.2 Jumps of stochastic processes 3
1.1.3 When are stochastic processes equal? 6
1.2 Measurability of Stochastic Processes 7
1.2.1 Filtration, adapted, and progressively measurable processes 8
1.2.2 Stopping times 13
1.2.3 Stopped variables, s-algebras, and truncated processes 19
1.2.4 Predictable processes 23
1.3 Martingales 29
1.3.1 Doob's inequalities 30
1.3.2 The energy equality 35
1.3.3 The quadratic variation of discrete time martingales 37
1.3.4 The downcrossings inequality 42
1.3.5 Regularization of martingales 46
1.3.6 The Optional Sampling Theorem 49
1.3.7 Application: elementary properties of Levy processes 58
1.3.8 Application: the first passage times of the Wiener processes 79
1.3.9 Some remarks on the usual assumptions 90
1.4 Localization 91
1.4.1 Stability under truncation 92
1.4.2 Local martingales 93
1.4.3 Convergence of local martingales: uniform convergence on compacts in probability 103
1.4.4 Locally bounded processes 105
2 Stochastic Integration with Locally Square-Integrable Martingales 107
2.1 The ItoStieltjes Integrals 108
2.1.1 ItoStieltjes integrals when the integrators have finite variation 110
2.1.2 ItoStieltjes integrals when the integrators are locally square-integrable martingales 116
2.1.3 ItoStieltjes integrals when the integrators are semimartingales 123
2.1.4 Properties of the ItoStieltjes integral 125
2.1.5 The integral process 125
2.1.6 Integration by parts and the existence of the quadratic variation 127
2.1.7 The KunitaWatanabe inequality 133
2.2 The Quadratic Variation of Continuous Local Martingales 137
2.3 Integration when Integrators are Continuous Semimartingales 145
2.3.1 The space of square-integrable continuous local martingales 146
2.3.2 Integration with respect to continuous local martingales 150
2.3.3 Integration with respect to semimartingales 161
2.3.4 The Dominated Convergence Theorem for stochastic integrals 161
2.3.5 Stochastic integration and the ItoStieltjes integral 163
2.4 Integration when Integrators are Locally Square-Integrable Martingales 166
2.4.1 The quadratic variation of locally square-integrable martingales 166
2.4.2 Integration when the integrators are locally square-integrable martingales 170
2.4.3 Stochastic integration when the integrators are semimartingales 175
3 The Structure of Local Martingales 178
3.1 Predictable Projection 181
3.1.1 Predictable stopping times 181
3.1.2 Decomposition of thin sets 187
3.1.3 The extended conditional expectation 189
3.1.4 Definition of the predictable projection 191
3.1.5 The uniqueness of the predictable projection, the predictable section theorem 193
3.1.6 Properties of the predictable projection 200
3.1.7 Predictable projection of local martingales 203
3.1.8 Existence of the predictable projection 205
3.2 Predictable Compensators 206
3.2.1 Predictable RadonNikodym Theorem 206
3.2.2 Predictable Compensator of locally integrable processes 212
3.2.3 Properties of the Predictable Compensator 216
3.3 The Fundamental Theorem of Local Martingales 218
3.4 Quadratic Variation 221
4 General Theory of Stochastic Integration 224
4.1 Purely Discontinuous Local Martingales 224
4.1.1 Orthogonality of local martingales 226
4.1.2 Decomposition of local martingales 231
4.1.3 Decomposition of semimartingales 233
4.2 Purely Discontinuous Local Martingales and Compensated Jumps 234
4.2.1 Construction of purely discontinuous local martingales 239
4.2.2 Quadratic variation of purely discontinuous local martingales 243
4.3 Stochastic Integration With Respect To Local Martingales 245
4.3.1 Definition of stochastic integration 247
4.3.2 Properties of stochastic integration 249
4.4 Stochastic Integration With Respect To Semimartingales 253
4.4.1 Integration with respect to special semimartingales 256
4.4.2 Linearity of the stochastic integral 260
4.4.3 The associativity rule 261
4.4.4 Change of measure 263
4.5 The Proof of Davis' Inequality 276
4.5.1 Discrete-time Davis' inequality 278
4.5.2 Burkholder's inequality 286
5 Some Other Theorems 291
5.1 The DoobMeyer Decomposition 291
5.1.1 The proof of the theorem 291
5.1.2 Dellacherie's formulas and the natural processes 297
5.1.3 The sub-super-and the quasi-martingales are semimartingales 301
5.2 Semimartingales as Good Integrators 306
5.3 Integration of Adapted Product Measurable Processes 313
5.4 Theorem of Fubini for Stochastic Integrals 317
5.5 Martingale Representation 326
6 Ito's formula 349
6.1 Ito's Formula for Continuous Semimartingales 351
6.2 Some Applications of the Formula 357
6.2.1 Zeros of Wiener processes 357
6.2.2 Continuous Levy processes 364
6.2.3 Levy's characterization of Wiener processes 366
6.2.4 Integral representation theorems for Wiener processes 371
6.2.5 Bessel processes 373
6.3 Change of measure for continuous semimartingales 375
6.3.1 Locally absolutely continuous change of measure 375
6.3.2 Semimartingales and change of measure 376
6.3.3 Change of measure for continuous semimartingales 378
6.3.4 Girsanov's formula for Wiener processes 380
6.3.5 KazamakiNovikov criteria 384
6.4 Ito's Formula for Non-Continuous Semimartingales 392
6.4.1 Ito's formula for processes with finite variation 396
6.4.2 The proof of Ito's formula 399
6.4.3 Exponential semimartingales 409
6.5 Ito's Formula For Convex Functions 415
6.5.1 Derivative of convex functions 416
6.5.2 Definition of local times 420
6.5.3 MeyerIto formula 427
6.5.4 Local times of continuous semimartingales 436
6.5.5 Local time of Wiener processes 443
6.5.6 RayKnight theorem 448
6.5.7 Theorem of Dvoretzky Erd.os and Kakutani 455
7 Processes with independent increments 458
7.1 Levy processes 458
7.1.1 Poisson processes 459
7.1.2 Compound Poisson processes generated by the jumps 462
7.1.3 Spectral measure of Levy processes 470
7.1.4 Decomposition of Levy processes 478
7.1.5 LevyKhintchine formula for Levy processes 484
7.1.6 Construction of Levy processes 487
7.1.7 Uniqueness of the representation 489
7.2 Predictable Compensators of Random Measures 494
7.2.1 Measurable random measures 495
7.2.2 Existence of predictable compensator 499
7.3 Characteristics of Semimartingales 506
7.4 LevyKhintchine Formula for Semimartingales with Independent Increments 511
7.4.1 Examples: probability of jumps of processes with independent increments 511
7.4.2 Predictable cumulants 516
7.4.3 Semimartingales with independent increments 521
7.4.4 Characteristics of semimartingales with independent increments 528
7.4.5 The proof of the formula 532
7.5 Decomposition of Processes with Independent Increments 536
Appendix 545
A Results from measure theory 545
A.1 The Monotone Class Theorem 545
A.2 Projection and the Measurable Selection Theorems 548
A.3 Cramer's Theorem 549
A.4 Interpretation of Stopped s-algebras 553
B Wiener processes 557
B.1 Basic Properties 557
B.2 Existence of Wiener Processes 565
B.3 Quadratic Variation of Wiener Processes 569
C Poisson processes 577
Notes and Comments 592
References 598
Index 601