Ruin probabilities and overshoots for general Lévy insurance risk processes

By Claudia Klüppelberg, Andreas E. Kyprianou, Ross Maller. 2004

196 citations recorded by the source corpus.

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.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/af0a9e3c24e89e18bd597e8531cf2b19f098e41a

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Citations in the corpus

Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory (2011, background): “…of actuarial mathematics, the first passage problem is very important to the ruin problems; see, for example, among others, Yang and Zhang (2001), Klüppelberg et al. (2004), Huzak et al. (2004a, 2004b), Zhou (2005), Garrido and Morales (2006), Biffis and Kyprianou (2010) and Billis and Morales…”
Occupation Times of Refracted Lévy Processes (2012, unknown): “See for example [1,4–6,8, 9,12,17,18].”
A Wiener–Hopf Monte Carlo simulation technique for Lévy processes (2009, background): “Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.”
WIENER-HOPF FACTORIZATION FOR LEVY PROCESSES HAVING POSITIVE JUMPS WITH RATIONAL TRANSFORMS (2008, background): “Among other works that study different aspects of ruin probabilities for Lévy processes, we mention [15], where asymptotic results for overshoots were obtained, [11], where a convolution formula was obtained for a classical claim process perturbed by an α-stable Lévy process, and [13], where a Polla”
Fluctuations of spectrally negative Markov additive processes (2008, background): “…of their fluctuation theory; see for example Schmidli (1999), Asmussen (2000), Asmussen and Kella (2000), Asmussen et al. (2004), Avram et al. (2004), Klüppelberg et al. (2004), Pistorius (2003), Huzak et al. (2004), Chan (2005), Chiu and Yin (2005) and Dube et al. (2004) to name but a few.”
Ratio of the Tail of an Infinitely Divisible Distribution on the Line to that of its Lévy Measure (2010, background): “See [32]. Applications of the classS( ) to Levy processes are found in [4; 6; 21 ].”
A Lévy Insurance Risk Process with Tax (2008, background): “This model has recently attracted a lot of research interest; see, e.g. [6], [8], [9], [11], [ 13 ], and [18].”
Ruin probabilities and decompositions for general perturbed risk processes (2004, background): “We end this Introduction by noting that in a very recent paper Klüppelberg, Kyprianou and Maller [7] study ruin probabilities for general Lévy insurance risk process (not necessarily spectrally negative) drifting to−∞.”
On The Expected Discounted Penalty function for Lévy Risk Processes (2006, methodology): “Ruin probabilities in this model have been discussed in Bertoin and Doney (1994), Yang and Zhang (2001), Morales and Schoutens (2003), Huzak et al. (2004), Klüppelberg et al. (2004) and in Doney and Kyprianou (2005).”
Distribution of the Present Value of Dividend Payments in a Lévy Risk Model (2007, background): “[11]: it is a Lévy process which drifts to infinity.”
Refracted Lévy processes (2008, background): “This preference is largely thanks to the robust mathematical theory which has been developed around certain path decompositions of such processes as well as the meaningful interpretation of the general spectrally negative Lévy process as an insurance risk process (see, for example, the discussion in”
Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process (2007, background): “Keywords: Reflected Lévy process; passage problem; integrated exponential Lévy process; insurance risk process; ruin 2000 Mathematics Subject Classification: Primary 60K05; 60K15; 91B30 Secondary 60G70; 60J55”
Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments (2005, background): “(Note that while the assumptions (ii) and (iii) are essentially technical, the assumption (i) is essential; in its absence we would need to pursue a different treatment—in the spirit of Kluppelberg, Kyprianou and Maller [ 24 ] and of Denisov, Foss and Korshunov [13].)”
Old and New Examples of Scale Functions for Spectrally Negative Levy Processes (2007, background): “Also using (33), the Laplace exponent of the decomposition (38) can be computed as follows with the help of an integration by parts; (κ+ λ− ζφ)θ + ζθ(2) − φθ ∫ ∞”
Convolution equivalence and distributions of random sums (2008, methodology): “An analogous problem for Levy processes is discussed in [ 18 ].”