Lévy Processes and Stochastic Calculus: Preface

Levy 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 Levy processes, then leading on to develop the stochastic calculus for Levy 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 Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs.

[1]  C L Menges,et al.  ON EINSTEIN'S THEORY OF RELATIVITY. , 1926, Science.

[2]  A. Kolmogoroff.  Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung , 1931 .

[3]  A. Wintner,et al.  Distribution functions and the Riemann zeta function , 1935 .

[4]  Helly.  Fourier transforms in the complex domain , 1936 .

[5]  Notes on linear transformations. I , 1936 .

[6]  S. Chandrasekhar.  Stochastic problems in Physics and Astronomy , 1943 .

[7]  R. Feynman.  Space-Time Approach to Non-Relativistic Quantum Mechanics , 1948 .

[8]  M. Kac.  On Some Connections between Probability Theory and Differential and Integral Equations , 1951 .

[9]  Kiyosi Itô.  On a formula concerning stochastic differentials , 1951 .

[10]  T. Teichmann,et al.  Harmonic Analysis and the Theory of Probability , 1957, The Mathematical Gazette.

[11]  Kiyosi Itô,et al.  SPECTRAL TYPE OF THE SHIFT TRANSFORMATION OF DIFFERENTIAL PROCESSES WITH STATIONARY INCREMENTS( , 1956 .

[12]  A. Beurling,et al.  DIRICHLET SPACES. , 1959, Proceedings of the National Academy of Sciences of the United States of America.

[13]  E. Hewitt,et al.  Theory of functions of a real variable , 1960 .

[14]  I. V. Girsanov.  On Transforming a Certain Class of Stochastic Processes by Absolutely Continuous Substitution of Measures , 1960 .

[15]  P. Meyer,et al.  A decomposition theorem for supermartingales , 1962 .

[16]  B. Mandlebrot.  The Variation of Certain Speculative Prices , 1963 .

[17]  P. Meyer.  Decomposition of supermartingales: The uniqueness theorem , 1963 .

[18]  Shinzo Watanabe,et al.  On a class of additive functionals of Markov processes , 1965 .

[19]  P. M. Lee.  Infinitely divisible stochastic processes , 1967 .

[20]  H. Kunita,et al.  On Square Integrable Martingales , 1967, Nagoya Mathematical Journal.

[21]  C. Doléans-Dade,et al.  Quelques applications de la formule de changement de variables pour les semimartingales , 1970 .

[22]  S. Port,et al.  Infinitely divisible processes and their potential theory. I , 1971 .

[23]  Bert Fristedt,et al.  Sample Functions of Stochastic Processes with Stationary, Independent Increments. , 1972 .

[24]  P. Millar.  Stochastic integrals and processes with stationary independent increments , 1972 .

[25]  $p$-variation de fonctions aléatoires. 2ième partie : processus à accroissements indépendants , 1972 .

[26]  K. Parthasarathy,et al.  Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory , 1972 .

[27]  Processus a Accroissements Independants , 1973 .

[28]  R. Mazo.  On the theory of brownian motion , 1973 .

[29]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[30]  P. Millar.  Exit properties of stochastic processes with stationary independent increments , 1973 .

[31]  Non-symmetric translation invariant Dirichlet forms , 1973 .

[32]  Takashi Komatsu,et al.  Markov processes associated with certain integro-differential operators , 1973 .

[33]  J. Jacod.  Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales , 1975 .

[34]  Paul-André Meyer,et al.  Questions de Theorie des Flots , 1975 .

[35]  Daniel W. Stroock,et al.  Diffusion processes associated with Lévy generators , 1975 .

[36]  R. C. Merton,et al.  Option pricing when underlying stock returns are discontinuous , 1976 .

[37]  Emil Grosswald,et al.  The student t-distribution of any degree of freedom is infinitely divisible , 1976 .

[38]  O. Barndorff-Nielsen.  Exponentially decreasing distributions for the logarithm of particle size , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[39]  I︠u︡. V. Linnik,et al.  Decomposition of Random Variables and Vectors , 1977 .

[40]  O. Barndorff-Nielsen,et al.  Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions , 1977 .

[41]  I. Monroe.  Processes that can be Embedded in Brownian Motion , 1978 .

[42]  Steven I. Marcus,et al.  Modeling and analysis of stochastic differential equations driven by point processes , 1978, IEEE Trans. Inf. Theory.

[43]  K. Parthasarathy.  Square integrable martingales orthogonal to every stochastic integral , 1978 .

[44]  C. Halgreen.  Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions , 1979 .

[45]  J. Jacod.  Calcul stochastique et problèmes de martingales , 1979 .

[46]  Charles M. Goldie,et al.  Subexponentiality and infinite divisibility , 1979 .

[47]  E. Davies,et al.  One-parameter semigroups , 1980 .

[48]  S. Wolfe.  On a continuous analogue of the stochastic difference equation Xn = ρ X n–1 + Bn , 1980, Advances in Applied Probability.

[49]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[50]  F. Knight.  Essentials of Brownian Motion and Diffusion , 1981 .

[51]  P. Embrechts,et al.  Comparing the Tail of an Infinitely Divisible Distribution with Integrals of its Levy Measure , 1981 .

[52]  Steven I. Marcus,et al.  Modeling and approximation of stochastic differential equations driven by semimartingales , 1981 .

[53]  Flot d'une équation différentielle stochastique , 1981 .

[54]  D. Aldous.  The Central Limit Theorem for Real and Banach Valued Random Variables , 1981 .

[55]  P. Hall.  A Comedy of Errors: The Canonical Form for a Stable Characteristic Function , 1981 .

[56]  E. Dynkin.  MARKOV PROCESSES AND RELATED PROBLEMS OF ANALYSIS , 1982 .

[57]  Ken-iti Sato,et al.  Stationary processes of ornstein-uhlenbeck type , 1983 .

[58]  E. Giné,et al.  THE CENTRAL LIMIT THEOREM FOR STOCHASTIC INTEGRALS WITH RESPECT TO LEVY PROCESSES , 1983 .

[59]  W. Vervaat,et al.  An integral representation for selfdecomposable banach space valued random variables , 1983 .

[60]  P. Baxendale,et al.  Brownian motions in the diffeomorphism group 1 , 1984 .

[61]  Tsukasa Fujiwara,et al.  Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group , 1985 .

[62]  J. Jacod.  Grossissement de filtration et processus d'Ornstein-Uhlenbeck generalise , 1985 .

[63]  Application de l'entropie métrique à la continuité des temps locaux des processus de Lévy , 1985 .

[64]  R. Léandre.  Flot d'une equation differentielle stochastique avec semi-martingale directrice discontinue , 1985 .

[65]  T. Ichinose,et al.  Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field , 1986 .

[66]  Jan Rosiński,et al.  On Ito Stochastic Integration with Respect to $p$-Stable Motion: Inner Clock, Integrability of Sample Paths, Double and Multiple Integrals , 1986 .

[67]  Anna Jnowska-Michalik.  On processes of ornstein-uhlenbeck type in hilbert space , 1987 .

[68]  S. Resnick.  Extreme Values, Regular Variation, and Point Processes , 1987 .

[69]  D. Elworthy.  Geometric aspects of diffusions on manifolds , 1988 .

[70]  Richard F. Bass,et al.  Uniqueness in law for pure jump Markov processes , 1988 .

[71]  G. Geoffrey Booth,et al.  The Stable-Law Model of Stock Returns , 1988 .

[72]  J. Leslie,et al.  On the non-closure under convolution of the subexponential family , 1989, Journal of Applied Probability.

[73]  R. R. Hall,et al.  AN INTRODUCTION TO THE THEORY OF THE RIEMANN ZETA‐FUNCTION , 1989 .

[74]  T. Ichinose,et al.  On essential selfadjointness of the Weyl quantized relativistic Hamiltonian , 1989 .

[75]  Distributions sur l'espace de P. Lévy et calcul stochastique , 1990 .

[76]  E. Seneta,et al.  The Variance Gamma (V.G.) Model for Share Market Returns , 1990 .

[77]  On solutions of stochastic differential equations with drift , 1990 .

[78]  N. Cutland,et al.  On homogeneous chaos , 1991, Mathematical Proceedings of the Cambridge Philosophical Society.

[79]  Stochastic differential equations of jump type on manifolds and Lévy flows , 1991 .

[80]  Peter Kuster,et al.  Malliavin calculus for processes with jumps , 1991 .

[81]  H. Föllmer,et al.  Hedging of contingent claims under incomplete in-formation , 1991 .

[82]  M. Marcus,et al.  Sample Path Properties of the Local Times of Strongly Symmetric Markov Processes Via Gaussian Processes , 1992 .

[83]  Xue-Mei Li.  Strong p-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds , 1994, 1911.07345.

[84]  M. Yor.  Some Aspects Of Brownian Motion , 1992 .

[85]  Peter H. Baxendale,et al.  Stability and Equilibrium Properties of Stochastic Flows of Diffeomorphisms , 1992 .

[86]  K. Parthasarathy.  An Introduction to Quantum Stochastic Calculus , 1992 .

[87]  M. Tsuchiya.  Lévy measure with generalized polar decomposition and the associated sde with jumps , 1992 .

[88]  根来 彬.  Stable-like processes : construction of the transition density and the behavior of sample paths near t=0 , 1993 .

[89]  M. Schürmann.  White Noise on Bialgebras , 1993 .

[90]  Etienne Pardoux,et al.  Stochastic partial differential equations, a review , 1993 .

[91]  Closability and resolvent of Dirichlet forms perturbed by jumps , 1993 .

[92]  David Applebaum,et al.  Lvy flows on manifolds and Lvy processes on Lie groups , 1993 .

[93]  Walter Hoh.  The martingale problem for a class of pseudo differential operators , 1994 .

[94]  N. Krylov.  Introduction to the theory of diffusion processes , 1994 .

[95]  The Inverse Gaussian Distribution: A Case Study in Exponential Families , 1994 .

[96]  H. Heyer,et al.  Harmonic Analysis of Probability Measures on Hypergroups , 1994 .

[97]  F. Delbaen,et al.  A general version of the fundamental theorem of asset pricing , 1994 .

[98]  M. Scheutzow,et al.  Perfect cocycles through stochastic differential equations , 1995 .

[99]  H. Walter.  Pseudo differential operators with negative definite symbols and the martingale problem , 1995 .

[100]  P. Lee,et al.  14. Simulation and Chaotic Behaviour of α‐Stable Stochastic Processes , 1995 .

[101]  A. Rodkina,et al.  Exponential stability of stochastic differential equations driven by discontinuous semimartingales , 1995 .

[102]  D. Nualart.  The Malliavin Calculus and Related Topics , 1995 .

[103]  Philip Protter,et al.  Stratonovich stochastic differential equations driven by general semimartingales , 1995 .

[104]  Hans U. Gerber,et al.  Option pricing by Esscher transforms. , 1995 .

[105]  Kai Lai Chung,et al.  From Brownian Motion To Schrödinger's Equation , 1995 .

[106]  Jean Picard,et al.  On the existence of smooth densities for jump processes , 1996 .

[107]  Michael M. Sørensen,et al.  A hyperbolic diffusion model for stock prices , 1996, Finance Stochastics.

[108]  R. Durrett.  Stochastic Calculus: A Practical Introduction , 1996 .

[109]  Stochastic differential equations with jumps and stochastic flows of diffeomorphisms , 1996 .

[110]  B. Øksendal.  AN INTRODUCTION TO MALLIAVIN CALCULUS WITH APPLICATIONS TO ECONOMICS , 1996 .

[111]  Philip Protter,et al.  The Euler scheme for Lévy driven stochastic differential equations , 1997 .

[112]  A. Weron,et al.  Approximation of stochastic differential equations driven by α-stable Lévy motion , 1997 .

[113]  Situ Rong.  On solutions of backward stochastic differential equations with jumps and applications , 1997 .

[114]  O. Barndorff-Nielsen.  Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling , 1997 .

[115]  P. A. Zanzotto.  On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion , 1997 .

[116]  René L. Schilling,et al.  Growth and Hölder conditions for the sample paths of Feller processes , 1998 .

[117]  From stochastic differential equation to quantum field theory , 1998, quant-ph/9810002.

[118]  Carl Mueller,et al.  The heat equation with Lévy noise1 , 1998 .

[119]  René L. Schilling,et al.  Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths , 1998 .

[120]  Erwan Saint Loubert Bié.  Etude d'une edps conduite par un bruit poissonnien , 1998 .

[121]  O. Barndorff-Nielsen,et al.  Some stationary processes in discrete and continuous time , 1998, Advances in Applied Probability.

[122]  S. Albeverio,et al.  Parabolic SPDEs driven by Poisson white noise , 1998 .

[123]  R. Schilling.  Subordination in the sense of Bochner and a related functional calculus , 1998, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[124]  Conservativeness and Extensions of Feller Semigroups , 1998 .

[125]  P. Carr,et al.  The Variance Gamma Process and Option Pricing , 1998 .

[126]  Terry Lyons.  Di erential equations driven by rough signals , 1998 .

[127]  Thomas Mikosch,et al.  Elementary stochastic calculus with finance in view , 1998 .

[128]  E. Eberlein,et al.  New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model , 1998 .

[129]  Elton P. Hsu.  Analysis on path and loop spaces , 1999 .

[130]  Financial Markets: Stochastic Analysis and the Pricing of Derivative Securities , 1999 .

[131]  J. Bertoin.  Subordinators: Examples and Applications , 1999 .

[132]  T. Chan.  Pricing contingent claims on stocks driven by Lévy processes , 1999 .

[133]  J. Pitman,et al.  Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions , 1999, math/9912170.

[134]  Ernst Eberlein,et al.  Term Structure Models Driven by General Lévy Processes , 1999 .

[135]  J. Rosenthal.  A First Look at Rigorous Probability Theory , 2000 .

[136]  David Applebaum,et al.  Compound Poisson Processes and Lévy Processes in Groups and Symmetric Spaces , 2000 .

[137]  A note on gamma random variables and Dirichlet series , 2000 .

[138]  Invariant measures for Lévy flows of diffeomorphisms , 2000, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[139]  T. Mikosch,et al.  Stochastic integral equations without probability , 2000 .

[140]  Vassili N. Kolokoltsov,et al.  Symmetric Stable Laws and Stable‐Like Jump‐Diffusions , 2000 .

[141]  Jan Kallsen,et al.  Optimal portfolios for logarithmic utility , 2000 .

[142]  V. Kolokoltsov.  Semiclassical Analysis for Diffusions and Stochastic Processes , 2000 .

[143]  R. Schilling,et al.  Feller semigroups, Lp -sub-Markovian semigroups, and applications to pseudo-differential operators with negative definite symbols , 2001 .

[144]  Dirichlet operators and the positive maximum principle , 2001 .

[145]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[146]  Sergei Levendorskii,et al.  Feller processes of normal inverse Gaussian type , 2001 .

[147]  THE INTERLACING CONSTRUCTION FOR STOCHASTIC FLOWS OF DIFFEOMORPHISMS ON EUCLIDEAN SPACES , 2001 .

[148]  Path-wise solutions of stochastic differential equations driven by Lévy processes , 2001 .

[149]  G. D. Lin,et al.  The Riemann zeta distribution , 2001 .

[150]  M. Yor,et al.  Stochastic Volatility for Levy Processes , 2001 .

[151]  N. H. Bingham,et al.  Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives , 2001 .

[152]  N. H. Bingham,et al.  Modelling asset returns with hyperbolic distributions , 2001 .

[153]  Ole E. Barndorff-Nielsen,et al.  Apparent scaling , 2001, Finance Stochastics.

[154]  Jia-An Yan,et al.  Introduction to Infinite Dimensional Stochastic Analysis , 2001 .

[155]  P. Protter.  A partial introduction to financial asset pricing theory , 2001 .

[156]  M. Yor,et al.  ASSET PRICES ARE BROWNIAN MOTION: ONLY IN BUSINESS TIME , 2001 .

[157]  J. Bertoin.  Some elements on Lévy processes , 2001 .

[158]  Stochastic flows of diffeomorphisms on manifolds driven by infinite-dimensional semimartingales with jumps , 2001 .

[159]  Marc Yor,et al.  Time Changes for Lévy Processes , 2001 .

[160]  Vector integration and stochastic integration in Banach spaces, by Nicolae Dinculeanu. Pp. 424. £58.50. 2000. ISBN 0 471 37738 4 (Wiley) , 2002, The Mathematical Gazette.

[161]  S. Levendorskii,et al.  Barrier options and touch- and-out options under regular Lévy processes of exponential type , 2002 .

[162]  A new path integral representation for the solutions of the Schrödinger, heat and stochastic Schrödinger equations , 2002, Mathematical Proceedings of the Cambridge Philosophical Society.

[163]  Jorge A. León,et al.  On Lévy processes, Malliavin calculus and market models with jumps , 2002, Finance Stochastics.

[164]  P. A. Meyer,et al.  Un Cours sur les Intégrales Stochastiques , 2002 .

[165]  N. Bingham,et al.  Semi-parametric modelling in finance: theoretical foundations , 2002 .

[166]  A. Etheridge.  A course in financial calculus , 2002 .

[167]  T. Chan,et al.  On the valuation of constant barrier options under spectrally one-sided exponential Lévy models and Carr's approximation for American puts , 2002 .

[168]  L. Mytnik.  Stochastic partial differential equation driven by stable noise , 2002 .

[169]  On stochastic differential equations driven by a Cauchy process and other stable Lévy motions , 2002 .

[170]  On Some Path Properties of Symmetric Stable-Like Processes for One Dimension , 2002 .

[171]  Jumping SDEs: absolute continuity using monotonicity , 2002 .

[172]  M. Yor,et al.  The Fine Structure of Asset Retums : An Empirical Investigation ' , 2006 .

[173]  M. Ryznar,et al.  Estimates of Green Function for Relativistic α-Stable Process , 2002 .

[174]  Raisa E. Feldman,et al.  Limit Distributions for Sums of Independent Random Vectors , 2002 .

[175]  J. Pitman,et al.  Self-similar processes with independent increments associated with Lévy and Bessel processes , 2002 .

[176]  J. Corcoran.  Modelling Extremal Events for Insurance and Finance , 2002 .

[177]  Yoshio Miyahara,et al.  The minimal entropy martingale measures for geometric Lévy processes , 2003, Finance Stochastics.

[178]  M. Meerschaert,et al.  Portfolio Modeling with Heavy Tailed Random Vectors , 2003 .

[179]  Stochastic partial differential equations driven by Lévy space-time white noise , 2004, math/0407131.

[180]  Sidney Resnick,et al.  On the foundations of multivariate heavy-tail analysis , 2004, Journal of Applied Probability.

[181]  Arne Løkka,et al.  Martingale Representation of Functionals of Lévy Processes , 2004 .

[182]  H. Kunita.  Representation of Martingales with Jumps and Applications to Mathematical Finance , 2004 .

[183]  Gennady Samorodnitsky,et al.  Stability of the trivial solution for linear stochastic differential equations with Poisson white noise , 2004 .

[184]  Hiroshi Kunita,et al.  Stochastic Differential Equations Based on Lévy Processes and Stochastic Flows of Diffeomorphisms , 2004 .

[185]  D. Stroock.  An Introduction to the Analysis of Paths on a Riemannian Manifold , 2005 .

[186]  Wim Schoutens,et al.  Exotic Option Pricing and Advanced Lévy Models , 2005 .

[187]  Extremal behavior of stochastic integrals driven by regularly varying Levy processes , 2007, math/0703802.

[188]  Anton Thalmaier,et al.  Stochastic Calculus of Variations in Mathematical Finance , 2005 .

[189]  R. C. Merton,et al.  Theory of Rational Option Pricing , 2015, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.

[190]  Zhi-Ming Ma,et al.  Extensions of Lévy–Khintchine formula and Beurling–Deny formula in semi-Dirichlet forms setting , 2006 .

[191]  Mark H. A. Davis,et al.  Malliavin Monte Carlo Greeks for jump diffusions , 2006 .

[192]  Peter Imkeller,et al.  First exit times of SDEs driven by stable Lévy processes , 2006 .

[193]  D. Applebaum.  Martingale-Valued Measures, Ornstein-Uhlenbeck Processes with Jumps and Operator Self-Decomposability in Hilbert Space , 2006 .

[194]  Henrik Hult,et al.  On regular variation for infinitely divisible random vectors and additive processes , 2006, Advances in Applied Probability.

[195]  Yasushi Ishikawa,et al.  Malliavin calculus on the Wiener–Poisson space and its application to canonical SDE with jumps , 2006 .