Book•
Lévy processes and infinitely divisible distributions
01 Jan 2013-
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.
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TL;DR: In this paper, the authors considered the tail asymptotic of discounted aggregate claims with compound dependence under risky investment, where the price of risky investment was modeled by a geometric Levy process, while claims were modeled by one-sided linear process whose innovations further obeying a so-called upper tail independence.
Abstract: This paper considers the tail asymptotic of discounted aggregate claims with compound dependence under risky investment. The price of risky investment is modeled by a geometric Levy process, while claims are modeled by a one-sided linear process whose innovations further obeying a so-called upper tail asymptotic independence. When the innovations are heavy tailed, we derive some uniform asymptotic formulas. The results show that the linear dependence has significant impact on the tail asymptotic of discounted aggregate claims but the upper tail asymptotic independence is negligible.
7 citations
Cites methods from "Lévy processes and infinitely divis..."
...For the general theory of Lévy processes, see, e.g. the monographs of Sato (1999) and Cont and Tankov (2004)....
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TL;DR: It is observed that with further decreases of α to small values α≪1, with λ≠0, locking between blue and red may be restored, and this nonmonotonic transition back to an ordered regime is surprising for a linear variation of a parameter such as the power law exponent.
Abstract: We examine a model of two interacting populations of phase oscillators labeled "blue" and "red." To this we apply tempered stable Levy noise, a generalization of Gaussian noise where the heaviness of the tails parametrized by a power law exponent α can be controlled by a tempering parameter λ. This system models competitive dynamics, where each population seeks both internal phase synchronization and a phase advantage with respect to the other population, subject to exogenous stochastic shocks. We study the system from an analytic and numerical point of view to understand how the phase lag values and the shape of the noise distribution can lead to steady or noisy behavior. Comparing the analytic and numerical studies shows that the bulk behavior of the system can be effectively described by dynamics in the presence of tilted ratchet potentials. Generally, changes in α away from the Gaussian noise limit 1<α<2 disrupt the locking between blue and red, while increasing λ acts to restore it. However, we observe that with further decreases of α to small values α≪1, with λ≠0, locking between blue and red may be restored. This is seen analytically in a restoration of metastability through the ratchet mechanism, and numerically in transitions between periodic and noisy regions in a fitness landscape using a measure of noise. This nonmonotonic transition back to an ordered regime is surprising for a linear variation of a parameter such as the power law exponent and provides a mechanism for guiding the collective behavior of such a complex competitive dynamical system.
7 citations
Cites background from "Lévy processes and infinitely divis..."
...,N; see [9] for a general introduction to Lévy processes....
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Posted Content•
TL;DR: For a L\'evy process on the real line, the authors provided complete criteria for the finiteness of exponential moments of the first passage time into the interval, the sojourn time in the interval and the last exit time from the interval.
Abstract: For a L\'evy process on the real line, we provide complete criteria for the finiteness of exponential moments of the first passage time into the interval $(r,\infty)$, the sojourn time in the interval $(-\infty,r]$, and the last exit time from $(-\infty,r]$. Moreover, whenever these quantities are finite, we derive their respective asymptotic behavior as $r \to \infty$.
7 citations
Cites background from "Lévy processes and infinitely divis..."
...7 in [15]) is equivalent to the distributional equalities Uq d = Vq + Wq for q > 0 where Vq and Wq are independent, Uq has the same distribution as Xτ with τ denoting an exponential random variable with parameter q independent of X , Vq has the same distribution as Sτ (with St := sup0≤s≤t Xs) and Wq has the same distribution as Iτ ....
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...17 in [15], this implies ∫ ∞ 1 eat t−1P{Xt < 0}dt = ∫ (1,∞) e at ν(dt) < ∞....
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Dissertation•
01 Jan 2014
TL;DR: In this article, the authors propose a model to accurately describe the price evolution of a risky asset, a security traded on a financial market such as a stock, currency or benchmark index.
Abstract: The role of the financial mathematician is to find solutions to problems in finance through the application of mathematical theory. The motivation for this work is specification of models to accurately describe the price evolution of a risky asset, a risky asset is for example a security traded on a financial market such as a stock, currency or benchmark index. This thesis makes contributions in two classes of models, namely activity time models and integer valued models, by the discovery of new real valued and integer valued stochastic processes. In both model frameworks applications to option pricing are considered.
7 citations
Cites background or methods or result from "Lévy processes and infinitely divis..."
...Since A1 = A2 = 0 and ν1(R), ν1(R) < ∞ from Sato (1999) theorem 12....
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...This section introduces some known results for Lévy processes, see for example Sato (1999). A cádlág stochastic process {L(t), t ≥ 0} on (Ω,F , {Ft},P) with values in R such that L(0) = 0 is called a Lévy process if...
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...(2.5.11) Furthermore from Theorem 3.6.6. in Jurek and Mason (1993) equation (2.5.11) is also equivalent to the condition stated in Sato (1999), namely∫ |x|>2 log |x|ν(dx) <∞ (2.5.12) where ν is the Lévy measure of the BDLP {Z(t), t ≥ 0}....
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...This section gives known results and definitions on Ornstein-Uhlenbeck processes, for complete details see Sato (1999), Barndorff-Nielsen, Jensen, and Sorensen (1998) and references therein....
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...This section introduces some known results for Lévy processes, see for example Sato (1999)....
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Posted Content•
TL;DR: In this article, the authors revisited the equity premium puzzle reported by Mehra and Prescott and showed that the large equity premium that they report can be explained by choosing a more appropriate distribution for the return data.
Abstract: In this paper, we revisit the equity premium puzzle reported in 1985 by Mehra and Prescott. We show that the large equity premium that they report can be explained by choosing a more appropriate distribution for the return data. We demonstrate that the high-risk aversion value observed by Mehra and Prescott may be attributable to the problem of fitting a proper distribution to the historical returns and partly caused by poorly fitting the tail of the return distribution. We describe a new distribution that better fits the return distribution and when used to describe historical returns can explain the large equity risk premium and thereby explains the puzzle.
7 citations
Cites background from "Lévy processes and infinitely divis..."
...In particular, if Bt is a Brownian motion, denoted by B 2 , with Lévy exponent, ψB(u) = − ln E [exp (iu B(1))] = −iuν + σ 2 2 u 2, and T(t), independent of B(t), is an IG4A Lévy subordinator is a Lévy process with increasing sample path (see Sato, 2002). 5See Chapter 6 of Sato (2002)....
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...In particular, if Bt is a Brownian motion, denoted by Bν,σ 2 , with Lévy exponent, ψB(u) = − ln E [exp (iu B(1))] = −iuν + σ 2 2 u 2, and T(t), independent of B(t), is an IG- 4A Lévy subordinator is a Lévy process with increasing sample path (see Sato, 2002)....
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...5See Chapter 6 of Sato (2002). subordinator with Laplace exponent given φT(1)(s) = − ln E [exp (−s T(1))] = − λT µT ©«1 − √ 1 + 2µ2T s λT ª®¬ , (17) then Y (t) = B(T(t)) is NIG process with Lévy exponent ψY (u) = φT (ψB(u)) = − λT µT ©«1 − √ 1 + 2µ2T λT ( −iuν + σ 2 2 u2 )ª®¬ ....
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References
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01 Jan 2014
TL;DR: In this article, Kloeden, P., Ombach, J., Cyganowski, S., Kostrikin, A. J., Reddy, J.A., Pokrovskii, A., Shafarevich, I.A.
Abstract: Algebra and Famous Inpossibilities Differential Systems Dumortier.: Qualitative Theory of Planar Jost, J.: Dynamical Systems. Examples of Complex Behaviour Jost, J.: Postmodern Analysis Jost, J.: Riemannian Geometry and Geometric Analysis Kac, V.; Cheung, P.: Quantum Calculus Kannan, R.; Krueger, C.K.: Advanced Analysis on the Real Line Kelly, P.; Matthews, G.: The NonEuclidean Hyperbolic Plane Kempf, G.: Complex Abelian Varieties and Theta Functions Kitchens, B. P.: Symbolic Dynamics Kloeden, P.; Ombach, J.; Cyganowski, S.: From Elementary Probability to Stochastic Differential Equations with MAPLE Kloeden, P. E.; Platen; E.; Schurz, H.: Numerical Solution of SDE Through Computer Experiments Kostrikin, A. I.: Introduction to Algebra Krasnoselskii, M.A.; Pokrovskii, A.V.: Systems with Hysteresis Kurzweil, H.; Stellmacher, B.: The Theory of Finite Groups. An Introduction Lang, S.: Introduction to Differentiable Manifolds Luecking, D.H., Rubel, L.A.: Complex Analysis. A Functional Analysis Approach Ma, Zhi-Ming; Roeckner, M.: Introduction to the Theory of (non-symmetric) Dirichlet Forms Mac Lane, S.; Moerdijk, I.: Sheaves in Geometry and Logic Marcus, D.A.: Number Fields Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis Matoušek, J.: Using the Borsuk-Ulam Theorem Matsuki, K.: Introduction to the Mori Program Mazzola, G.; Milmeister G.; Weissman J.: Comprehensive Mathematics for Computer Scientists 1 Mazzola, G.; Milmeister G.; Weissman J.: Comprehensive Mathematics for Computer Scientists 2 Mc Carthy, P. J.: Introduction to Arithmetical Functions McCrimmon, K.: A Taste of Jordan Algebras Meyer, R.M.: Essential Mathematics for Applied Field Meyer-Nieberg, P.: Banach Lattices Mikosch, T.: Non-Life Insurance Mathematics Mines, R.; Richman, F.; Ruitenburg, W.: A Course in Constructive Algebra Moise, E. E.: Introductory Problem Courses in Analysis and Topology Montesinos-Amilibia, J.M.: Classical Tessellations and Three Manifolds Morris, P.: Introduction to Game Theory Nikulin, V.V.; Shafarevich, I. R.: Geometries and Groups Oden, J. J.; Reddy, J. N.: Variational Methods in Theoretical Mechanics Øksendal, B.: Stochastic Differential Equations Øksendal, B.; Sulem, A.: Applied Stochastic Control of Jump Diffusions Poizat, B.: A Course in Model Theory Polster, B.: A Geometrical Picture Book Porter, J. R.; Woods, R.G.: Extensions and Absolutes of Hausdorff Spaces Radjavi, H.; Rosenthal, P.: Simultaneous Triangularization Ramsay, A.; Richtmeyer, R.D.: Introduction to Hyperbolic Geometry Rees, E.G.: Notes on Geometry Reisel, R. B.: Elementary Theory of Metric Spaces Rey, W. J. J.: Introduction to Robust and Quasi-Robust Statistical Methods Ribenboim, P.: Classical Theory of Algebraic Numbers Rickart, C. E.: Natural Function Algebras Roger G.: Analysis II Rotman, J. J.: Galois Theory Jost, J.: Compact Riemann Surfaces Applications ́ Introductory Lectures on Fluctuations of Levy Processes with Kyprianou, A. : Rautenberg, W.; A Concise Introduction to Mathematical Logic Samelson, H.: Notes on Lie Algebras Schiff, J. L.: Normal Families Sengupta, J.K.: Optimal Decisions under Uncertainty Séroul, R.: Programming for Mathematicians Seydel, R.: Tools for Computational Finance Shafarevich, I. R.: Discourses on Algebra Shapiro, J. H.: Composition Operators and Classical Function Theory Simonnet, M.: Measures and Probabilities Smith, K. E.; Kahanpää, L.; Kekäläinen, P.; Traves, W.: An Invitation to Algebraic Geometry Smith, K.T.: Power Series from a Computational Point of View Smoryński, C.: Logical Number Theory I. An Introduction Stichtenoth, H.: Algebraic Function Fields and Codes Stillwell, J.: Geometry of Surfaces Stroock, D.W.: An Introduction to the Theory of Large Deviations Sunder, V. S.: An Invitation to von Neumann Algebras Tamme, G.: Introduction to Étale Cohomology Tondeur, P.: Foliations on Riemannian Manifolds Toth, G.: Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems Wong, M.W.: Weyl Transforms Xambó-Descamps, S.: Block Error-Correcting Codes Zaanen, A.C.: Continuity, Integration and Fourier Theory Zhang, F.: Matrix Theory Zong, C.: Sphere Packings Zong, C.: Strange Phenomena in Convex and Discrete Geometry Zorich, V.A.: Mathematical Analysis I Zorich, V.A.: Mathematical Analysis II Rybakowski, K. P.: The Homotopy Index and Partial Differential Equations Sagan, H.: Space-Filling Curves Ruiz-Tolosa, J. R.; Castillo E.: From Vectors to Tensors Runde, V.: A Taste of Topology Rubel, L.A.: Entire and Meromorphic Functions Weintraub, S.H.: Galois Theory
401 citations
TL;DR: In this article, several definitions of the Riesz fractional Laplace operator in R^d have been studied, including singular integrals, semigroups of operators, Bochner's subordination, and harmonic extensions.
Abstract: This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner's subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain.
372 citations
TL;DR: In this article, the authors give an up-to-date account of the theory and applications of scale functions for spectrally negative Levy processes, including the first extensive overview of how to work numerically with scale functions.
Abstract: The purpose of this review article is to give an up to date account of the theory and applications of scale functions for spectrally negative Levy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Levy processes, in particular a reasonable understanding of the Levy–Khintchine formula and its relationship to the Levy–Ito decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Levy processes; (Bertoin, Levy Processes (1996); Sato, Levy Processes and Infinitely Divisible Distributions (1999); Kyprianou, Introductory Lectures on Fluctuations of Levy Processes and Their Applications (2006); Doney, Fluctuation Theory for Levy Processes (2007)), Applebaum Levy Processes and Stochastic Calculus (2009).
288 citations
TL;DR: A closed formula for prices of perpetual American call options in terms of the overall supremum of the Lévy process, and a corresponding closed formulas for perpetual American put options involving the infimum of the after-mentioned process are obtained.
Abstract: Consider a model of a financial market with a stock driven by a Levy process and constant interest rate. A closed formula for prices of perpetual American call options in terms of the overall supremum of the Levy process, and a corresponding closed formula for perpetual American put options involving the infimum of the after-mentioned process are obtained. As a direct application of the previous results, a Black-Scholes type formula is given. Also as a consequence, simple explicit formulas for prices of call options are obtained for a Levy process with positive mixed-exponential and arbitrary negative jumps. In the case of put options, similar simple formulas are obtained under the condition of negative mixed-exponential and arbitrary positive jumps. Risk-neutral valuation is discussed and a simple jump-diffusion model is chosen to illustrate the results.
269 citations
01 May 2013
TL;DR: In this paper, the authors review work on extreme events, their causes and consequences, by a group of European and American researchers involved in a three-year project on these topics.
Abstract: We review work on extreme events, their causes and consequences, by a group of European and American researchers involved in a three-year project on these topics. The review covers theoretical aspects of time series analysis and of extreme value theory, as well as of the deterministic modeling of extreme events, via continuous and discrete dynamic models. The applications include climatic, seismic and socio-economic events, along with their prediction. Two important results refer to (i) the complementarity of spectral analysis of a time series in terms of the continuous and the discrete part of its power spectrum; and (ii) the need for coupled modeling of natural and socio-economic systems. Both these results have implications for the study and prediction of natural hazards and their human impacts.
166 citations