but there are better books out there on stochastic calculus
and Levy processes. The material covered is essentially a rewriting of existing mathematics. There are also minor math mistakes throughout. For example on page 197, the definition
of stochastic integration and the definition of random
measures on page 89 are consistent for defining stochastic
integration with respect to Brownian motion only if we
we assume Brownian motion is a function of finite variation which is not.
Author(s): David Applebaum
Series: Cambridge Studies in Advanced Mathematics 116
Edition: 2
Publisher: Cambridge University Press
Year: 2009
Language: English
Pages: 492
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 11
Preface to Second Edition......Page 15
Preface......Page 17
Overview......Page 23
Notation......Page 31
1.1 Review of measure and probability......Page 33
1.1.1 Measure and probability spaces......Page 34
1.1.2 Random variables, integration and expectation......Page 37
1.1.3 Conditional expectation......Page 41
1.1.4 Independence and product measures......Page 44
1.1.5 Convergence of random variables......Page 46
1.1.6 Characteristic functions......Page 48
1.1.7 Stochastic processes......Page 51
1.2.1 Convolution of measures......Page 53
1.2.2 Definition of infinite divisibility......Page 56
1.2.3 Examples of in.nite divisibility......Page 58
1.2.4 The Lévy–Khintchine formula......Page 60
1.2.5 Stable random variables......Page 65
The Riemann zeta distribution......Page 71
A relativistic distribution......Page 72
1.3 Lévy processes......Page 75
1.3.1 Examples of Lévy processes......Page 78
1.3.2 Subordinators......Page 84
1.4 Convolution semigroups of probability measures......Page 94
1.4.1 Canonical Lévy processes......Page 96
1.5 Some further directions in Lévy processes......Page 99
1.5.1 Recurrence and transience......Page 100
1.5.2 Wiener–Hopf factorisation......Page 101
1.5.3 Local times......Page 102
1.5.4 Regular Variation and Subexponentiality......Page 103
1.6 Notes and further reading......Page 110
1.7 Appendix: An exercise in calculus......Page 112
2 Martingales, stopping times and random measures......Page 114
2.1.1 Filtrations and adapted processes......Page 115
2.1.2 Martingales and Lévy processes......Page 116
2.1.3 Martingale spaces......Page 122
2.2 Stopping times......Page 123
2.2.1 The Doob–Meyer decomposition......Page 125
2.2.2 Stopping times and Lévy processes......Page 127
2.3 The jumps of a Lévy process – Poisson randommeasures......Page 131
2.3.1 Random measures......Page 135
2.3.2 Poisson integration......Page 138
2.3.3 Processes of finite variation......Page 142
2.4 The Lévy–Itô decomposition......Page 144
2.5 Moments of Lévy Processes......Page 163
2.6.1 Limit events – a review......Page 165
2.6.2 Interlacing......Page 166
2.7 Semimartingales......Page 169
2.8 Notes and further reading......Page 170
2.9 Appendix: càdlàg functions......Page 171
2.10 Appendix: Unitary action of the shift......Page 173
3.1.1 Markov processes and transition functions......Page 175
3.1.2 Sub-Markov processes......Page 184
3.2 Semigroups and their generators......Page 185
3.3 Semigroups and generators of Lévy processes......Page 192
3.3.1 Translation-invariant semigroups......Page 193
3.3.2 Representation of semigroups and generators by pseudo-differential operators......Page 195
3.4.1 Lp-Markov semigroups and Lévy processes......Page 204
3.4.2 Self-adjoint semigroups......Page 208
3.5.1 The positive maximum principle and Courrège’s theorems......Page 212
3.5.2 Examples of Lévy-type operators......Page 217
3.5.3 The forward equation......Page 220
3.6.1 Dirichlet forms and sub-Markov semigroups......Page 221
3.6.3 Closable Markovian forms......Page 224
3.6.4 Dirichlet forms and Hunt processes......Page 227
3.6.5 Non-symmetric Dirichlet forms......Page 229
3.7 Notes and further reading......Page 232
3.8.1 Basic concepts: operators, domains, closure, graphs, cores, resolvents......Page 233
3.8.2 Dual and adjoint operators – self-adjointness......Page 238
3.8.3 Closed symmetric forms......Page 240
3.8.4 The Fourier transform and pseudo-differential operators......Page 242
4.1 Integrators and integrands......Page 246
4.2.1 The L2-theory......Page 253
4.2.2 The extended theory......Page 257
4.3.1 Brownian stochastic integrals......Page 261
4.3.2 Poisson stochastic integrals......Page 262
4.3.3 Lévy-type stochastic integrals......Page 265
4.3.5 Wiener–Lévy integrals, moving averages and the Ornstein–Uhlenbeck process......Page 269
4.4.4 Applications of Itô’s Formula......Page 275
4.4.2 Itô’s formula for Lévy-type stochastic integrals......Page 281
4.4.3 Quadratic variation and Itô’s product formula......Page 289
Lévy’s characterisation of Brownian motion......Page 293
Burkholder’s Inequality......Page 294
Moments of Lévy-type stochastic integrals – Kunita’s inequalities......Page 297
Poisson random measures revisited......Page 300
The Stratonovitch integral......Page 302
The Marcus canonical integral......Page 304
4.4.6 Backwards stochastic integrals......Page 306
4.4.7 Local times and extensions of Itô’s formula......Page 310
4.5 Notes and further reading......Page 311
5 Exponential martingales, change of measure and financial applications......Page 312
5.1 Stochastic exponentials......Page 313
5.2.1 Lévy-type stochastic integrals as local martingales......Page 317
5.2.2 Exponential martingales......Page 319
5.2.3 Change of measure – Girsanov’s theorem......Page 324
The Cameron–Martin–Maruyama theorem......Page 327
5.3 Martingale representation theorems......Page 331
5.4.1 Orientation......Page 338
5.4.2 Symmetric Functions......Page 339
5.4.3 Construction of Multiple Wiener–It Integrals......Page 341
Iterated stochastic integrals......Page 343
5.4.4 The Chaos decomposition......Page 345
5.5 Introduction to Malliavin Calculus......Page 349
5.6 Stochastic calculus and mathematical finance......Page 356
5.6.1 Introduction to financial derivatives......Page 357
Portfolios......Page 359
5.6.2 Stock prices as a Lévy process......Page 361
5.6.3 Change of measure......Page 363
5.6.4 The Black–Scholes formula......Page 365
5.6.5 Incomplete markets......Page 369
5.6.6 The generalised Black–Scholes equation......Page 371
Hyperbolic distributions......Page 373
Option pricing with hyperbolic Lévy processes......Page 375
5.6.8 Other Lévy process models for stock prices......Page 378
5.8 Appendix: Bessel functions......Page 380
5.9 Appendix: A density result......Page 383
6 Stochastic differential equations......Page 386
6.1 Differential equations and flows......Page 387
6.2 Stochastic differential equations – existence and uniqueness......Page 395
SDEs driven by Lévy processes......Page 409
Stochastic exponentials......Page 410
Diffusion processes......Page 412
6.4.1 Stochastic flows......Page 415
6.4.2 The Markov property......Page 419
6.4.3 Cocycles......Page 421
6.5 Interlacing for solutions of SDEs......Page 424
6.6 Continuity of solution flows to SDEs......Page 427
6.7.1 SDEs and Feller semigroups......Page 431
6.7.2 The Feynman–Kac formula......Page 437
6.7.3 Weak solutions to SDEs and the martingale problem......Page 441
6.8 Lyapunov exponents for stochastic differential equations......Page 443
6.9 Densities for Solutions of SDEs......Page 447
6.10 Marcus canonical equations......Page 449
6.11 Notes and further reading......Page 458
References......Page 463
Index of notation......Page 481
Subject Index......Page 486
