LÃ©vy Processes and Stochastic Calculus

L_vy processes form a wide and rich class of random process, and have many applications ranging from physics to finance. Stochastic calculus is the mathematics of systems interacting with random noise. Here, the author ties these two subjects together, beginning with an introduction to the general theory of L_vy processes, then leading on to develop the stochastic calculus for L_vy processes in a direct and accessible way. This fully revised edition now features a number of new topics. These include: regular variation and subexponential distributions; necessary and sufficient conditions for L_vy processes to have finite moments; characterization of L_vy processes with finite variation; Kunita's estimates for moments of L_vy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general L_vy processes; multiple Wiener-L_vy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for L_vy-driven SDEs.

Preface to Second Edition	p. xiii
Preface	p. xv
Overview	p. xxi
Notation	p. xxix
Lévy processes	p. 1
Review of measure and probability	p. 1
Infinite divisibility	p. 21
Lévy processes	p. 43
Convolution semigroups of probability measures	p. 62
Some further directions in Lévy processes	p. 67
Notes and further reading	p. 78
Appendix: An exercise in calculus	p. 80
Martingales, stopping times and random measures	p. 82
Martingales	p. 83
Stopping times	p. 83
The jumps of a Lévy process - Poisson random measures	p. 99
The Lévy-Itô decomposition	p. 112
Moments of Lévy Processes	p. 131
The interlacing construction	p. 133
Semimartingales	p. 137
Notes and further reading	p. 138
Appendix: càdlàg functions	p. 139
Appendix: Unitary action of the shift	p. 141
Markov processes, semigroups and generators	p. 143
Markov processes, evolutions and semigroups	p. 143
Semigroups and their generators	p. 153
Semigroups and generators of Lévy processes	p. 160
LP-Markov semigroups	p. 172
Lévy-type operators and the positive maximum principle	p. 180
Dirichlet forms	p. 189
Notes and further reading	p. 200
Appendix: Unbounded operators in Banach spaces	p. 201
Stochastic integration	p. 214
Integrators and integrands	p. 214
Stochastic integration	p. 221
Stochastic integrals based on Lévy processes	p. 229
Itô's formula	p. 243
Notes and further reading	p. 279
Exponential martingales, change of measure and financial applications	p. 280
Stochastic exponentials	p. 281
Exponential martingales	p. 285
Martingale representation theorems	p. 299
Multiple Wiener-Lévy Integrals	p. 306
Introduction to Malliavin Calculus	p. 317
Stochastic calculus and mathematical finance	p. 324
Notes and further reading	p. 348
Appendix: Bessel functions	p. 348
Appendix: A density result	p. 351
Stochastic differential equations	p. 354
Differential equations and flows	p. 355
Stochastic differential equations - existence and uniqueness	p. 363
Examples of SDEs	p. 377
Stochastic flows, cocycle and Markov properties of SDEs	p. 383
Interlacing for solutions of SDEs	p. 392
Continuity of solution flows to SDEs	p. 395
Solutions of SDEs as Feller processes, the Feynman-Kac formula and martingale problems	p. 399
Lyapunov exponents for stochastic differential equations	p. 411
Densities for Solutions of SDEs	p. 415
Marcus canonical equations	p. 417
Notes and further reading	p. 426
References	p. 431
Index of notation	p. 449
Subject index	p. 454
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LÃ©vy Processes and Stochastic Calculus

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