A Note on Scale Functions and the Time Value of Ruin for Lévy Insurance Risk Processes

We examine discounted penalties at ruin for surplus dynamics driven by a general spectrally negative Levy process; the natural class of stochastic processes which contains many examples of risk processes which have already been considered in the existing literature. Following from the important contributions of [Zhou, X., 2005. On a classical risk model with a constant dividend barrier. North Am. Act. J. 95-108] we provide an explicit characterization of a generalized version of the Gerber-Shiu function in terms of scale functions, streamlining and extending results available in the literature.

[1]  A note on r-balayages of matrix-exponential L'evy processes , 2009 .

[2]  Hansjörg Furrer,et al.  Risk processes perturbed by α-stable Lévy motion , 1998 .

[3]  Z. Palmowski,et al.  Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process , 2007, Journal of Applied Probability.

[4]  S. Taylor,et al.  LÉVY PROCESSES (Cambridge Tracts in Mathematics 121) , 1998 .

[5]  Manuel Morales.  On the expected discounted penalty function for a perturbed risk process driven by a subordinator , 2007 .

[6]  Andreas E. Kyprianou,et al.  General tax Structures and the Lévy Insurance Risk Model , 2009, Journal of Applied Probability.

[7]  Hans U. Gerber,et al.  On the discounted penalty at ruin in a jump-diffusion and the perpetual put option , 1998 .

[8]  David C. M. Dickson,et al.  On the distribution of the surplus prior to ruin , 1992 .

[9]  On the moments of the surplus process perturbed by diffusion , 2002 .

[10]  A. Kyprianou,et al.  Some explicit identities associated with positive self-similar Markov processes. , 2007, 0708.2383.

[11]  R. Loeffen,et al.  An Optimal Dividends Problem with a Terminal Value for Spectrally Negative Lévy Processes with a Completely Monotone Jump Density , 2009, Journal of Applied Probability.

[12]  Jun Cai,et al.  On the expected discounted penalty function at ruin of a surplus process with interest , 2002 .

[13]  R. Wolpert.  Lévy Processes , 2000 .

[14]  The quintuple law for sums of dependent Levy processes , 2007 .

[15]  A. Kyprianou,et al.  Old and New Examples of Scale Functions for Spectrally Negative Levy Processes , 2007, 0801.0393.

[16]  A. E. Kyprianou,et al.  Overshoots and undershoots of Lèvy processes , 2006 .

[17]  Miljenko Huzak,et al.  Ruin probabilities and decompositions for general perturbed risk processes , 2004, math/0407125.

[18]  Last Exit Before an Exponential Time for Spectrally Negative Lévy Processes , 2009, Journal of Applied Probability.

[19]  H. Gerber,et al.  A Note on the Dividends-Penalty Identity and the Optimal Dividend Barrier , 2006, ASTIN Bulletin.

[20]  Chuancun Yin,et al.  Optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes: An alternative approach , 2009, J. Comput. Appl. Math..

[21]  Andreas E. Kyprianou,et al.  Refracted Lévy processes , 2008 .

[22]  Aleksandr Alekseevich Borovkov,et al.  Stochastic processes in queueing theory , 1976 .

[23]  A. Salimi Shamloo,et al.  Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions , 2008 .

[24]  M. Caballero,et al.  Conditioned stable Lévy processes and the Lamperti representation , 2006, Journal of Applied Probability.

[25]  Xiaowen Zhou,et al.  On a Classical Risk Model with a Constant Dividend Barrier , 2005 .

[26]  Hans U. Gerber,et al.  The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin , 1997 .

[27]  Exact and asymptotic n-tuple laws at first and last passage , 2008, 0811.3075.

[28]  Mladen Savov,et al.  Smoothness of scale functions for spectrally negative Lévy processes , 2009, 0903.1467.

[29]  Lothar Breuer,et al.  A quintuple law for Markov additive processes with phase-type jumps , 2010, Journal of Applied Probability.

[30]  Ronnie Loeffen,et al.  An optimal dividends problem with transaction costs for spectrally negative Lévy processes , 2009 .

[31]  R. L. Loeen.  An optimal dividends problem with a terminal value for spectrally negative Levy processes with a completely monotone jump density , 2008 .

[32]  Xiaowen Zhou,et al.  Distribution of the Present Value of Dividend Payments in a Lévy Risk Model , 2007, Journal of Applied Probability.

[33]  Hansjörg Albrecher,et al.  A Lévy Insurance Risk Process with Tax , 2008, Journal of Applied Probability.

[34]  H. Gerber,et al.  On the Time Value of Ruin , 1997 .

[35]  D. Dickson.  On the distribution of the claim causing ruin , 1993 .

[36]  Roger Lee.  Option Pricing by Transform Methods: Extensions, Unification, and Error Control , 2004 .

[37]  Sung Nok Chiu,et al.  Passage times for a spectrally negative Lévy process with applications to risk theory , 2005 .

[38]  José Garrido,et al.  On The Expected Discounted Penalty function for Lévy Risk Processes , 2006 .

[39]  E. Biffis,et al.  On a generalization of the Gerber–Shiu function to path-dependent penalties☆ , 2010 .

[40]  Amaury Lambert,et al.  Completely asymmetric Lévy processes confined in a finite interval , 2000 .

[41]  L. C. G. Rogers.  Evaluating first-passage probabilities for spectrally one-sided Lévy processes , 2000, Journal of Applied Probability.

[42]  Hailiang Yang,et al.  Spectrally negative Lévy processes with applications in risk theory , 2001, Advances in Applied Probability.

[43]  A. Kyprianou.  Introductory Lectures on Fluctuations of Lévy Processes with Applications , 2006 .

[44]  Jun Cai,et al.  Ruin probabilities and penalty functions with stochastic rates of interest , 2004 .

[45]  R. Loeffen,et al.  On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes , 2008, 0811.1862.

[46]  P. Patie.  Law of the exponential functional of a new family of one-sided Levy processes via self-similar continuous state branching processes with immigration and the Wright hypergeometric functions , 2007, 0712.1115.

[47]  D. Duffie,et al.  Transform Analysis and Asset Pricing for Affine Jump-Diffusions , 1999 .

[48]  Gordon E. Willmot,et al.  Analysis of a defective renewal equation arising in ruin theory , 1999 .

[49]  P. Patie.  Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration , 2009 .

[50]  B. Surya.  Evaluating Scale Functions of Spectrally Negative Lévy Processes , 2008 .

[51]  A Potential-theoretical Review of some Exit Problems of Spectrally Negative Lévy Processes , 2005 .

[52]  R. Song,et al.  Convexity and Smoothness of Scale Functions and de Finetti’s Control Problem , 2008, 0801.1951.

[53]  An optimal dividends problem with transaction costs for spectrally negative Levy processes , 2008 .

[54]  P. Kythe,et al.  Computational Methods for Linear Integral Equations , 2002 .