Andreas E. Kyprianou

Ruin probabilities and overshoots for general Lévy insurance risk processes

We formulate the insurance risk process in a general Levy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to −∞ a.s. and the positive tail of the Levy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Kluppelberg [Stochastic Process. Appl. 64 (1996) 103–125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207–226] for ruin probabilities and the overshoot in random walk and compound Poisson models are shown to have analogues in the general setup. The identities we derive open the way to further investigation of general renewal-type properties of Levy processes.

The Theory of Scale Functions for Spectrally Negative Lévy Processes

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).

Some remarks on first passage of Levy processes, the American put and pasting principles

The purpose of this article is to provide, with the help of a fluctuation identity, a generic link between a number of known identities for the first passage time and overshoot above/below a fixed level of a Levy process and the solution of Gerber and Shiu [Astin Bull. 24 (1994) 195–220], Boyarchenko and Levendorskii [Working paper series EERS 98/02 (1998), Unpublished manuscript (1999), SIAM J. Control Optim. 40 (2002) 1663–1696], Chan [Original unpublished manuscript (2000)], Avram, Chan and Usabel [Stochastic Process. Appl. 100 (2002) 75–107], Mordecki [Finance Stoch. 6 (2002) 473–493], Asmussen, Avram and Pistorius [Stochastic Process. Appl. 109 (2004) 79–111] and Chesney and Jeanblanc [Appl. Math. Fin. 11 (2004) 207–225] to the American perpetual put optimal stopping problem. Furthermore, we make folklore precise and give necessary and sufficient conditions for smooth pasting to occur in the considered problem.

Overshoots and undershoots of Lèvy processes

We obtain a new fluctuation identity for a general Levy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Kluppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766–1801] concerning asymptotic overshoot distribution of a particular class of Levy processes with semi-heavy tails and refine some of their main conclusions. In particular, we explain how different types of first passage contribute to the form of the asymptotic overshoot distribution established in the aforementioned paper. Applications in insurance mathematics are noted with emphasis on the case that the underlying Levy process is spectrally one sided

Some calculations for Israeli options

Abstract.Recently Kifer (2000) introduced the concept of an Israeli (or Game) option. That is a general American-type option with the added possibility that the writer may terminate the contract early inducing a payment exceeding the holder’s claim had they exercised at that moment. Kifer shows that pricing and hedging of these options reduces to evaluating a saddle point problem associated with Dynkin games. In this short text we give two examples of perpetual Israeli options where the solutions are explicit.

Fluctuations of Lévy Processes with Applications

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

Meromorphic Lévy processes and their fluctuation identities

The last couple of years has seen a remarkable number of new, explicit examples of the Wiener.Hopf factorization for Levy processes where previously there had been very few. We mention, in particular, the many cases of spectrally negative Levy processes in [Sixth Seminar on Stochastic Analysis, Random Fields and Applications (2011) 119.146, Electron. J. Probab. 13 (2008) 1672.1701], hyper-exponential and generalized hyper-exponential Levy processes [Quant. Finance 10 (2010) 629.644], Lamperti-stable processes in [J. Appl. Probab. 43 (2006) 967.983, Probab. Math. Statist. 30 (2010) 1.28, Stochastic Process. Appl. 119 (2009) 980.1000, Bull. Sci. Math. 133 (2009) 355.382], Hypergeometric processes in [Ann. Appl. Probab. 20 (2010) 522.564, Ann. Appl. Probab. 21 (2011) 2171.2190, Bernoulli 17 (2011) 34.59], β-processes in [Ann. Appl. Probab. 20 (2010) 1801.1830] and θ-processes in [J. Appl. Probab. 47 (2010) 1023.1033]. In this paper we introduce a new family of Levy processes, which we call Meromorphic Levy processes, or just M-processes for short, which overlaps with many of the aforementioned classes. A key feature of the M-class is the identification of theirWiener.Hopf factors as rational functions of infinite degree written in terms of poles and roots of the Laplace exponent, all of which are real numbers. The specific structure of the M-class Wiener.Hopf factorization enables us to explicitly handle a comprehensive suite of fluctuation identities that concern first passage problems for finite and infinite intervals for both the process itself as well as the resulting process when it is reflected in its infimum. Such identities are of fundamental interest given their repeated occurrence in various fields of applied probability such as mathematical finance, insurance risk theory and queuing theory.

Refracted Lévy processes

Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Levy processes. The latter is a Levy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted Levy process is described by the unique strong solution to the stochastic differential equation dU(t) = -delta 1({Ut > b})dt + dX(t), where X = {X-t: t >= 0) is a Levy process with law P and b, delta is an element of R such that the resulting process U may visit the half line (b, infinity) with positive probability. We consider in particular the case that X is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the q-scale function of the driving Levy process and its perturbed version describing motion above the level b. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.

Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels

We revisit the previous work of Leland [J Finance 49:1213–1252, 1994], Leland and Toft [J Finance 51:987–1019, 1996] and Hilberink and Rogers [Finance Stoch 6:237–263, 2002] on optimal capital structure and show that the issue of determining an optimal endogenous bankruptcy level can be dealt with analytically and numerically when the underlying source of randomness is replaced by that of a general spectrally negative Lévy process. By working with the latter class of processes we bring to light a new phenomenon, namely that, depending on the nature of the small jumps, the optimal bankruptcy level may be determined by a principle of continuous fit as opposed to the usual smooth fit. Moreover, we are able to prove the optimality of the bankruptcy level according to the appropriate choice of fit.

A Martingale Review of some Fluctuation Theory for Spectrally Negative Lévy Processes

We give a review of some fluctuation theory for spectrally negative Levy processes using for the most part martingale theory. The methodology is based on the techniques found in Kyprianou and Palmowski (2003) which deal with similar issues for a general class of Markov additive processes.