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.
Citations
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TL;DR: In this article, the authors studied the convergence rate of the weak Euler approximation for solutions to Levy-driven stochastic differential equations with non-egenerate main part driven by a spherically symmetrized sparsification.
Abstract: This paper studies the rate of convergence of the weak Euler approximation for solutions to Levy-driven stochastic differential equations with nondegenerate main part driven by a spherically symmet...
6 citations
Cites background from "Lévy processes and infinitely divis..."
...In particular, Lévy processes [1], [4], [24] are the simplest generic class of processes having a....
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Dissertation•
29 Jun 2012
TL;DR: In this article, the authors study the problem of portfolio insurance from the point of view of a fund manager, who guarantees to the investor that the portfolio value at maturity will be above a fixed threshold.
Abstract: In this thesis I'm interested in two aspects of portfolio management: the portfolio insurance under a risk measure constraint and quadratic hedge in incomplete markets. Part I. I study the problem of portfolio insurance from the point of view of a fund manager, who guarantees to the investor that the portfolio value at maturity will be above a fixed threshold. If, at maturity, the portfolio value is below the guaranteed level, a third party will refund the investor up to the guarantee. In exchange for this protection, for which the investor pays a given fee, the third party imposes a limit on the risk exposure of the fund manager, in the form of a convex monetary risk measure. The fund manager therefore tries to maximize the investor's utility function subject to the risk measure constraint. I give a full solution to this nonconvex optimization problem in the complete market setting and show in particular that the choice of the risk measure is crucial for the optimal portfolio to exist. An interesting outcome is that the fund manager's maximization problem may not admit an optimal solution for all convex risk measures, which means that not all convex risk measures may be used to limit fund's exposure in this way. I provide conditions for the existence of the solution. Explicit results are provided for the entropic risk measure (for which the optimal portfolio always exists), for the class of spectral risk measures (for which the optimal portfolio may fail to exist in some cases) and for the G-divergence. Key words: Portfolio Insurance; utility maximization; convex risk measure; VaR, CVaR and spectral risk measure; entropy and G-divergence. Part II. In the second part I study the problem of quadratic hedge in incomplete markets. I work with a three-dimensional Markov jump process: the first component is the state variable representing the hedging instrument traded in the market, the second component models a risk factor which "perturbs" the dynamics of the hedging instrument and is not traded in the market (as a volatility factor for example in stochastic volatility models); the third one is another source of risk which affects the option's payoff at maturity and is also not traded in the market. The problem can be seen then as a constrained quadratic hedge problem. I privilege here the dynamic programming approach which allows me to obtain the HJB equation related to the value function. This equation is semi linear and non local due the presence of jumps. The main result of this thesis is that this value function, as a function of the initial wealth, is a second order polynomial whose coefficients are characterized as the unique smooth solutions of a triplet of PIDEs, the first of which is semi linear and does not depend on the particular choice of option one wants to hedge, the other two being simply linear. This result is stated when the Markov process is assumed to be a non-generate jump-diffusion and when it is a pure jump process. I finally apply my theoretical results to an example of quadratic hedge in the context of electricity markets.
6 citations
Cites background from "Lévy processes and infinitely divis..."
...For more details see for example Triebel (1992); Gilbarg and Trudinger (2001); Adams and Fournier (2009)....
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...…:= E [exp(iwLt)] := exp(tl(w)) and it is a well known result that l(w) = −σα|w|α ( 1− iβsign(ω) tan πα 2 + icω ) (E.1) where σ := [ − ( g(0+) + g(0−) ) Γ(−α) cos (πα 2 )]1/α β := g(0+)− g(0−) g(0+) + g(0−) See for example Proposition 28.3 in Sato (1999) or Section 3.7 in Cont and Tankov (2004)....
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Dissertation•
01 Jan 2014
TL;DR: The results show various circumstances in which normally diffusing chemokines fail to recruit adequate phagocytes and more importantly this behaviour stays the same even if the source of chemokine production is multiplied by several orders of magnitude.
Abstract: Chemotaxis is the major cytotaxic mechanism that leads the movement of phagocytes in the tissue towards the harmful agents. Loss of phagocytes ability to track and respond to danger signals can lead to chronic infections, sepsis or even death. This thesis examines the consequences of anomalous diffusion of chemokines on the chemotaxis of phagocytes in the event of acute inflammatory responses. The main driver of any chemotactic system is the corresponding chemo-attractant, which is the role given to chemokines. Allowing anomalous (fractional) diffusion with the tail index of 0 < α < 2, leads to the front propagation rate proportional to t: faster than the traditional Gaussian spread (t). Moreover, fractional chemokine concentration profiles obey power laws, which results in slower tail decays leading to heavy tails; whereas in the Gaussian scenario tail decays are exponential and rapid. Changing the morphology of chemokine profile over the domain will affect all other entities that depend on chemokine concentration: the likes of tactic motility, sensitivity and velocity. Our study aims at understanding the influences of chemokine gradient field variations on phagocyte chemotaxis and hence on the acute inflammatory response. We show various circumstances in which normally diffusing chemokines fail to recruit adequate phagocytes and more importantly this behaviour stays the same even if the source of chemokine production is multiplied by several orders of magnitude. Another challenge is to insure the presence of an optimum number of phagocytes in the tissue, which is governed by a timely initiation of infiltration.
6 citations
Cites background from "Lévy processes and infinitely divis..."
...One of the most important applications of subordinators in stochastic processes is in ‘time changing’ [5, 161]....
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...Majority of the material in this chapter can be found in standard probability and functional analysis textbooks such as [5, 12, 46, 52, 53, 117, 147, 161, 204]....
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01 Jan 2012
TL;DR: It is demonstrated that monkeys shift their behaviors in a systematic way, and that they do so in a near optimal manner, which gives further evidence that the DDM is a good model for both the behavior and neural activity related to perceptual choice.
Abstract: The research presented in this thesis is a collection of applications and extensions of stochastic accumulator models to various areas of decision making and attention in neuroscience. Ch. 1 introduces the major techniques and experimental results that guide us throughout the rest of the thesis. In particular, we introduce and define the leaky, competing accumulator, drift diffusion, and Ornstein-Uhlenbeck models. In Ch. 2, we adopt an Ornstein-Uhlenbeck (OU) process to fit a generalized version of the motion dots task in which monkeys are now faced with biased rewards. We demonstrate that monkeys shift their behaviors in a systematic way, and that they do so in a near optimal manner. We also fit the OU model to neural data and find that OU model behaves almost like a pure drift diffusion process. This gives further evidence that the DDM is a good model for both the behavior and neural activity related to perceptual choice. In Ch. 3, we construct a multi-area model for a covert search task. We discover some new trends in the data and systematically construct a model which explains the key findings in the data. Our model proposes that the lateral intraparietal area (LIP) plays an attentional role in this covert search task, and suggests that the two monkeys used in this study adapted different strategies for performing the task. In Ch. 4, we extend the model of noise in the popular drift diffusion model (DDM) to a more general Lévy process. The jumps introduced into the noise increments dramatically affect the reaction times predicted by the DDM, and they allow the pure DDM to reproduce fast error trials given unbiased initial data, a feature which other models require more parameters to reproduce. The model is fit to human subject data and is shown to outperform the extended DDM in data containing fast error reaction times. In Ch. 5, we construct a model for studying capacity constraints on cognitive
6 citations
TL;DR: In this paper, the authors analyze the Bayesian formulation of the sequential testing of two simple hypotheses for the distributional characteristics of an inverse Gaussian process and show that the initial optimal stopping problem for the posterior probability of one of the hypotheses can be reduced to a free-boundary problem.
Abstract: We analyze the Bayesian formulation of the sequential testing of two simple hypotheses for the distributional characteristics of an inverse Gaussian process. This problem arises when we are willing to test the positive drift of an unobservable Brownian motion, for which only the first passage times over positive thresholds can be recorded. We show that the initial optimal stopping problem for the posterior probability of one of the hypotheses can be reduced to a free-boundary problem, whose unknown boundary points are characterized by the principles of the continuous or smooth fit and whose unknown value function solves a linear integro-differential equation over the continuation set. A numerical scheme, based on the collocation method for boundary value problems, is further illustrated, in order to get precise approximations of the free-boundary problem solution, which seems to be very hard to derive analytically, because of the particular structure of the Levy measure of an inverse Gaussian process.
6 citations
Cites background from "Lévy processes and infinitely divis..."
...The following properties of X are easily inferred from (2.1) by means of standard arguments (see, e.g., Sato, 1999, ch. 2 and 4): (1) X is a purely jump process; (2) X has in nite jump activity, in the sense that for any t > 0 it has in nitely many jumps on (0, t); (3) X is a subordinator, meaning…...
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...…be the likelihood ratio process de ned by ϕt := d(P1|FXt )/d(P0|FXt ); standard results on the density transformation for Lévy processes (see, e.g., Sato, 1999, Theorem 33.2, p. 219) imply that ϕt = exp ( 1 2 (µ20 − µ 2 1)Xt − t ∫ ∞ 0 ( e(µ 2 0−µ21)x/2 − 1 ) e−µ20x/2 √ 2πx3 dx ) = exp ( 1 2…...
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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