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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; characterisation 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 --- Stochastic analysis --- Lévy, Processus de --- Analyse stochastique --- Lévy processes. --- Stochastic analysis. --- Lévy processes --- Lévy, Processus de --- Lévy processes. --- Stochastic integral equations. --- Integral equations --- Random walks (Mathematics) --- Analysis, Stochastic --- Mathematical analysis --- Stochastic processes --- Levy processes.

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The theory of Lévy processes in Lie groups is not merely an extension of the theory of Lévy processes in Euclidean spaces. Because of the unique structures possessed by non-commutative Lie groups, these processes exhibit certain interesting limiting properties which are not present for their counterparts in Euclidean spaces. These properties reveal a deep connection between the behaviour of the stochastic processes and the underlying algebraic and geometric structures of the Lie groups themselves. The purpose of this work is to provide an introduction to Lévy processes in general Lie groups, the limiting properties of Lévy processes in semi-simple Lie groups of non-compact type and the dynamical behavior of such processes as stochastic flows on certain homogeneous spaces. The reader is assumed to be familiar with Lie groups and stochastic analysis, but no prior knowledge of semi-simple Lie groups is required.

Lévy processes. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Random walks (Mathematics) --- Lévy processes --- Lie groups --- Levy processes.

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Recent years have seen an explosion of interest in stochastic partial differential equations where the driving noise is discontinuous. In this comprehensive monograph, two leading experts detail the evolution equation approach to their solution. Most of the results appeared here for the first time in book form. The authors start with a detailed analysis of Lévy processes in infinite dimensions and their reproducing kernel Hilbert spaces; cylindrical Lévy processes are constructed in terms of Poisson random measures; stochastic integrals are introduced. Stochastic parabolic and hyperbolic equations on domains of arbitrary dimensions are studied, and applications to statistical and fluid mechanics and to finance are also investigated. Ideal for researchers and graduate students in stochastic processes and partial differential equations, this self-contained text will also interest those working on stochastic modeling in finance, statistical physics and environmental science.

Stochastic partial differential equations --- Lévy processes --- Stochastic partial differential equations. --- Lévy processes. --- Équations aux dérivées partielles stochastiques --- Lévy, Processus de --- Lévy processes --- Équations aux dérivées partielles stochastiques --- Lévy, Processus de --- Lévy processes. --- Random walks (Mathematics) --- Banach spaces, Stochastic differential equations in --- Hilbert spaces, Stochastic differential equations in --- SPDE (Differential equations) --- Stochastic differential equations in Banach spaces --- Stochastic differential equations in Hilbert spaces --- Differential equations, Partial --- Levy processes.

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The Lévy Laplacian is an infinite-dimensional generalization of the well-known classical Laplacian. The theory has become well developed in recent years and this book was the first systematic treatment of the Lévy-Laplace operator. The book describes the infinite-dimensional analogues of finite-dimensional results, and more especially those features which appear only in the generalized context. It develops a theory of operators generated by the Lévy Laplacian and the symmetrized Lévy Laplacian, as well as a theory of linear and nonlinear equations involving it. There are many problems leading to equations with Lévy Laplacians and to Lévy-Laplace operators, for example superconductivity theory, the theory of control systems, the Gauss random field theory, and the Yang-Mills equation. The book is complemented by an exhaustive bibliography. The result is a work that will be valued by those working in functional analysis, partial differential equations and probability theory.

Functional analysis --- Laplacian operator. --- Lévy processes. --- Harmonic functions. --- Functions, Harmonic --- Laplace's equations --- Bessel functions --- Differential equations, Partial --- Fourier series --- Harmonic analysis --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Random walks (Mathematics) --- Operator, Laplacian --- Levy processes.

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Exit problems for one-dimensional Lévy processes are easier when jumps only occur in one direction. In the last few years, this intuition became more precise: we know now that a wide variety of identities for exit problems of spectrally-negative Lévy processes may be ergonomically expressed in terms of two q-harmonic functions (or scale functions or positive martingales) W and Z. The proofs typically require not much more than the strong Markov property, which hold, in principle, for the wider class of spectrally-negative strong Markov processes. This has been established already in particular cases, such as random walks, Markov additive processes, Lévy processes with omega-state-dependent killing, and certain Lévy processes with state dependent drift, and seems to be true for general strong Markov processes, subject to technical conditions. However, computing the functions W and Z is still an open problem outside the Lévy and diffusion classes, even for the simplest risk models with state-dependent parameters (say, Ornstein–Uhlenbeck or Feller branching diffusion with phase-type jumps).

Lévy processes --- non-random overshoots --- skip-free random walks --- fluctuation theory --- scale functions --- capital surplus process --- dividend payment --- optimal control --- capital injection constraint --- spectrally negative Lévy processes --- reflected Lévy processes --- first passage --- drawdown process --- spectrally negative process --- dividends --- de Finetti valuation objective --- variational problem --- stochastic control --- optimal dividends --- Parisian ruin --- log-convexity --- barrier strategies --- adjustment coefficient --- logarithmic asymptotics --- quadratic programming problem --- ruin probability --- two-dimensional Brownian motion --- spectrally negative Lévy process --- general tax structure --- first crossing time --- joint Laplace transform --- potential measure --- Laplace transform --- first hitting time --- diffusion-type process --- running maximum and minimum processes --- boundary-value problem --- normal reflection --- Sparre Andersen model --- heavy tails --- completely monotone distributions --- error bounds --- hyperexponential distribution --- reflected Brownian motion --- linear diffusions --- drawdown --- Segerdahl process --- affine coefficients --- spectrally negative Markov process --- hypergeometric functions --- capital injections --- bankruptcy --- reflection and absorption --- Pollaczek–Khinchine formula --- scale function --- Padé approximations --- Laguerre series --- Tricomi–Weeks Laplace inversion