1 L´evy processes 1
1.1 Review of measure and probability 1
1.2 Infinite divisibility 20
1.3 L´evy processes 39
1.4 Convolution semigroups of probability measures 58
1.5 Some further directions in L´evy processes 63
1.6 Notes and further reading 67
1.7 Appendix: An exercise in calculus 69
2 Martingales, stopping times and random measures 70
2.1 Martingales 70
2.2 Stopping times 78
2.3 The jumps of a L´evy process – Poisson random measures 86
2.4 The L´evy–Itˆo decomposition 96
2.5 The interlacing construction 111
2.6 Semimartingales 115
2.7 Notes and further reading 116
2.8 Appendix: c`adl`ag functions 117
3 Markov processes, semigroups and generators 120
3.1 Markov processes, evolutions and semigroups 120
3.2 Semigroups and their generators 129
3.3 Semigroups and generators of L´evy processes 136
3.4 Lp-Markov semigroups 148
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viii Contents
3.5 L´evy-type operators and the positive maximum principle 156
3.6 Dirichlet forms 164
3.7 Notes and further reading 176
3.8 Appendix: Unbounded operators in Banach spaces 176
4Stochastic integration 190
4.1 Integrators and integrands 190
4.2 Stochastic integration 197
4.3 Stochastic integrals based on L´evy processes 205
4.4 Itˆo’s formula 218
4.5 Notes and further reading 245
5 Exponential martingales, change of measure and financial
applications 246
5.1 Stochastic exponentials 247
5.2 Exponential martingales 250
5.3 Martingale representation theorems 262
5.4 Stochastic calculus and mathematical finance 267
5.5 Notes and further reading 288
5.6 Appendix: Bessel functions 289
6 Stochastic differential equations 292
6.1 Differential equations and flows 293
6.2 Stochastic differential equations – existence and uniqueness 301
6.3 Examples of SDEs 314
6.4 Stochastic flows, cocycle and Markov properties of SDEs 319
6.5 Interlacing for solutions of SDEs 328
6.6 Continuity of solution flows to SDEs 331
6.7 Solutions of SDEs as Feller processes, the Feynman–Kac
formula and martingale problems 337
6.8 Marcus canonical equations 348
6.9 Notes and further reading 357
References 360
Index of notation 375
Subject index