Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory

We consider the spectrally negative Lévy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum, and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Furthermore, a new expression for the generalized Dickson’s formula is provided.

[1]  V. Zolotarev.  The First Passage Time of a Level and the Behavior at Infinity for a Class of Processes with Independent Increments , 1964 .

[2]  D. J. Emery.  Exit problem for a spectrally positive process , 1973, Advances in Applied Probability.

[3]  N. H. Bingham,et al.  Fluctuation theory in continuous time , 1975, Advances in Applied Probability.

[4]  Alfredo D. Egídio dos Reis,et al.  How long is the surplus below zero , 1993 .

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

[6]  Jean Bertoin,et al.  Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval , 1997 .

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

[8]  T. Rolski.  Stochastic Processes for Insurance and Finance , 1999 .

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

[10]  Tomasz Rolski,et al.  Stochastic Processes for Insurance and Finance , 2001 .

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

[12]  Total duration of negative surplus for the compound Poisson process that is perturbed by diffusion , 2002, Journal of Applied Probability.

[13]  The joint density function of three characteristics on jump-diffusion risk process , 2003 .

[14]  On the distribution of duration of stay in an interval of the semi-continuous process with independent increments , 2004 .

[15]  Claudia Kluppelberg,et al.  Ruin probabilities and overshoots for general Lévy insurance risk processes , 2004 .

[16]  Some fluctuation identities for Lévy processes with jumps of the same sign , 2004 .

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

[18]  Florin Avram,et al.  Exit problems for spectrally negative Levy processes and applications to (Canadized) Russian options , 2004 .

[19]  Miljenko Huzak,et al.  Ruin probabilities for competing claim processes , 2004, Journal of Applied Probability.

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

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

[22]  Zbigniew Palmowski,et al.  A Martingale Review of some Fluctuation Theory for Spectrally Negative Lévy Processes , 2005 .

[23]  Andreas E. Kyprianou,et al.  Some remarks on first passage of Levy processes, the American put and pasting principles , 2005 .

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

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

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

[27]  On extreme ruinous behaviour of Lévy insurance risk processes , 2006, Journal of Applied Probability.

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

[29]  Florin Avram,et al.  On the optimal dividend problem for a spectrally negative Lévy process , 2007, math/0702893.

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

[31]  R. Doney,et al.  Fluctuation Theory for Lévy Processes , 2007 .

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

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

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

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

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

[37]  Claudia Kluppelberg,et al.  The first passage event for sums of dependent Lévy processes with applications to insurance risk. , 2009 .

[38]  Andreas E. Kyprianou,et al.  A Note on Scale Functions and the Time Value of Ruin for Lévy Insurance Risk Processes , 2009 .

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

[40]  Xiaowen Zhou,et al.  OCCUPATION TIMES OF SPECTRALLY NEGATIVE LÉVY PROCESSES WITH APPLICATIONS , 2010, 1012.3448.

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

[42]  A. Kyprianou.  Fluctuations of Lévy Processes with Applications , 2014 .