Applebaum D. Lévy Processes and Stochastic Calculus

Second Edition. — Cambridge University Press, 2009. — 460 p. — ISBN: 978-0521738651.The aim of this book is to provide a straightforward and accessible introduction
to stochastic integrals and stochastic differential equations driven by Lévy
processes.
Lévy 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 Lévy processes, then leading on to develop the stochastic calculus for Lévy 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 Lévy processes to have finite moments; characterization of Lévy processes with finite variation; Kunita's estimates for moments of Lévy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Lévy processes; multiple Wiener-Lévy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Lévy-driven SDEs.Lévy processes.
Review of measure and probability.
Infinite divisibility.
Lévy processes.
Convolution semigroups of probability measures.
Some further directions in Lévy processes.
Notes and further reading.
Appendix: An exercise in calculus.
Martingales, stopping times and random measures.
Martingales.
Stopping times.
The jumps of a Lévy process – Poisson random measures.
The Lévy–Itô decomposition.
Moments of Lévy Processes.
The interlacing construction.
Semimartingales.
Notes and further reading.
Appendix: càdlàg functions.
Appendix: Unitary action of the shift.
Markov processes, semigroups and generators.
Markov processes, evolutions and semigroups.
Semigroups and their generators.
Semigroups and generators of Lévy processes.
L^p-Markov semigroups.
Lévy-type operators and the positive maximum principle.
Dirichlet forms.
Notes and further reading.
Appendix: Unbounded operators in Banach spaces.
Stochastic integration.
Integrators and integrands.
Stochastic integration.
Stochastic integrals based on Lévy processes.
Itô’s formula.
Notes and further reading.
Exponential martingales, change of measure and financial applications.
Stochastic exponentials.
Exponential martingales.
Martingale representation theorems.
Multiple Wiener–Lévy Integrals.
Introduction to Malliavin Calculus.
Stochastic calculus and mathematical finance.
Notes and further reading.
Appendix: Bessel functions.
Appendix: A density result.
Stochastic differential equations.
Differential equations and flows.
Stochastic differential equations – existence and uniqueness.
Examples of SDEs.
Stochastic flows,cycle and Markov properties of SDEs.
Interlacing for solutions of SDEs.
Continuity of solution flows to SDEs.
Solutions of SDEs as Feller processes, the Feynman–Kac formula and martingale problems.
Lyapunov exponents for stochastic differential equations.
Densities for Solutions of SDEs.
Marcus canonical equations.
Notes and further reading.Index of notation.
Subject index.