Cover -- Half-title -- Series-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Overview -- Notation -- 1 Lévy processes -- 1.1 Review of measure and probability -- 1.1.1 Measure and probability spaces -- 1.1.2 Random variables, integration and expectation -- 1.1.3 Conditional expectation -- 1.1.4 Independence and product measures -- 1.1.5 Convergence of random variables -- 1.1.6 Characteristic functions -- 1.1.7 Stochastic processes -- 1.1.8 Random fields -- 1.2 Infinite divisibility -- 1.2.1 Convolution of measures -- 1.2.2 Definition of infinite divisibility -- 1.2.3 Examples of infinite divisibility -- 1.2.4 The Lévy-Khintchine formula -- Notes -- 1.2.5 Stable random variables -- 1.2.6 Diversion: Number theory and relativity -- 1.3 Lévy processes -- 1.3.1 Examples of Lévy processes -- 1.3.2 Subordinators -- 1.4 Convolution semigroups of probability measures -- 1.4.1 Canonical Lévy processes -- 1.4.2 Modification of Lévy processes -- 1.5 Some further directions in Lévy processes -- 1.5.1 Recurrence and transience -- 1.5.2 Wiener-Hopf factorisation -- 1.5.3 Local times -- 1.6 Notes and further reading -- 1.7 Appendix: An exercise in calculus -- 2 Martingales, stopping times and random measures -- 2.1 Martingales -- 2.1.1 Filtrations and adapted processes -- 2.1.2 Martingales and Lévy processes -- 2.1.3 Martingale spaces -- 2.2 Stopping times -- 2.2.1 The Doob-Meyer decomposition -- 2.2.2 Stopping times and Lévy processes -- 2.3 The jumps of a Lévy process - Poisson random measures -- 2.3.1 Random measures -- 2.3.2 Poisson integration -- 2.3.3 Processes of finite variation -- 2.4 The Lévy-It decomposition -- 2.5 The interlacing construction -- 2.5.1 Limit events - a review -- 2.5.2 Interlacing -- 2.6 Semimartingales -- 2.7 Notes and further reading -- 2.8 Appendix: càdlàg functions.

3 Markov processes, semigroups and generators -- 3.1 Markov processes, evolutions and semigroups -- 3.1.1 Markov processes and transition functions -- 3.1.2 Sub-Markov processes -- 3.2 Semigroups and their generators -- 3.3 Semigroups and generators of Lévy processes -- 3.3.1 Translation-invariant semigroups -- 3.3.2 Representation of semigroups and generators by pseudo-differential operators -- 3.3.3 Subordination of semigroups -- 3.4 Lp-Markov semigroups -- 3.4.1 Lp-Markov semigroups and Lévy processes -- 3.4.2 Self-adjoint semigroups -- 3.5 Lévy-type operators and the positive maximum principle -- 3.5.1 The positive maximum principle and Courrège's theorems -- 3.5.2 Examples of Lévy-type operators -- 3.5.3 The forward equation -- 3.6 Dirichlet forms -- 3.6.1 Dirichlet forms and sub-Markov semigroups -- 3.6.2 The Beurling-Deny formula -- 3.6.3 Closable Markovian forms -- 3.6.4 Dirichlet forms and Hunt processes -- 3.6.5 Non-symmetric Dirichlet forms -- 3.7 Notes and further reading -- 3.8 Appendix: Unbounded operators in Banach spaces -- 3.8.1 Basic concepts: operators domains, closure, graphs, cores, resolvents -- 3.8.2 Dual and adjoint operators - self-adjointness -- 3.8.3 Closed symmetric forms -- 3.8.4 The Fourier transform and pseudo-differential operators -- 4 Stochastic integration -- 4.1 Integrators and integrands -- 4.2 Stochastic integration -- 4.2.1 The L2-theory -- 4.2.2 The extended theory -- 4.3 Stochastic integrals based on Lévy processes -- 4.3.1 Brownian stochastic integrals -- 4.3.2 Poisson stochastic integrals -- 4.3.3 Lévy-type stochastic integrals -- 4.3.4 Stable stochastic integrals -- 4.3.5 Wiener-Lévy integrals, moving averages and the Ornstein-Uhlenbeck process -- 4.4 Itô's formula -- 4.4.1 Itô's formula for Brownian integrals -- 4.4.2 Itô's formula for Lévy-type stochastic integrals.

4.4.3 Quadratic variation and Itô's product formula -- 4.4.4 The Stratonovitch and Marcus canonical integrals -- The Stratonovitch integral -- The Marcus canonical integral -- 4.4.5 Backwards stochastic integrals -- 4.4.6 Local times and extensions of Itô's formula -- 4.5 Notes and further reading -- 5 Exponential martingales, change of measure and financial applications -- 5.1 Stochastic exponentials -- 5.2 Exponential martingales -- 5.2.1 Lévy-type stochastic integrals as local martingales -- 5.2.2 Exponential martingales -- 5.2.3 Change of measure - Girsanov's theorem -- 5.2.4 Analysis on Wiener space -- The Cameron-Martin-Maruyama theorem -- Directional derivative and integration by parts -- 5.3 Martingale representation theorems -- 5.4 Stochastic calculus and mathematical finance -- 5.4.1 Introduction to financial derivatives -- Portfolios -- 5.4.2 Stock prices as a Lévy process -- 5.4.3 Change of measure -- 5.4.4 The Black-Scholes formula -- 5.4.5 Incomplete markets -- The Föllmer-Schweizer minimal measure -- The Esscher transform -- 5.4.6 Hyperbolic Lévy processes in finance -- Hyperbolic distributions -- Option pricing with hyperbolic Lévy processes -- 5.4.7 Other Lévy process models for stock prices -- 5.5 Notes and further reading -- 5.6 Appendix: Bessel functions -- 6 Stochastic differential equations -- 6.1 Differential equations and flows -- 6.2 Stochastic differential equations - existence and uniqueness -- 6.3 Examples of SDEs -- SDEs driven by Lévy processes -- Stochastic exponentials -- The Langevin equation and Ornstein-Uhlenbeck process revisited -- Diffusion processes -- Jump-diffusion processes -- 6.4 Stochastic flows, cocycle and Markov properties of SDEs -- 6.4.1 Stochastic flows -- 6.4.2 The Markov property -- 6.4.3 Cocycles -- 6.5 Interlacing for solutions of SDEs -- 6.6 Continuity of solution flows to SDEs.

6.7 Solutions of SDEs as Feller processes, the Feynman-Kac formula and martingale problems -- 6.7.1 SDEs and Feller semigroups -- 6.7.2 The Feynman-Kac formula -- 6.7.3 Weak solutions to SDEs and the martingale problem -- 6.8 Marcus canonical equations -- 6.9 Notes and further reading -- References -- Index of notation -- Subject index.