Overshoots and undershoots of Lèvy processes

By R. Doney, Andreas E. Kyprianou. 2006

147 citations recorded by the source corpus.

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

Subfields: Finance, Management Information Systems, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/b7d705905bbad6ff259162f3cab4e040bf26e822

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Citations in the corpus

Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory (2011, methodology): “For example, Doney and Kyprianou (2006) studied the problem for general Lévy process X .”
Convolution equivalence and distributions of random sums (2008, background): “370 T. Watanabe by [6,14,18,24,29,34]. See also [ 7 ].”
Special, conjugate and complete scale functions for spectrally negative Lévy processes (2007, background): “For example, in the theory of optimal stopping [1; 2] and [38], in the theory of optimal control [3], [40; 51] and [44], in the theory of queuing and storage models [21] and [4], in the theory of branching processes [10], [29] and [43], in the theory of insurance risk and ruin [15], [34], [33], [20]”
Convexity and Smoothness of Scale Functions and de Finetti’s Control Problem (2008, background): “of literature concerning actuarial mathematics which explores the interaction of classical models of risk and fine properties of Lévy processes with a view to gaining new results on both sides (see for example [2, 10, 18, 19, 23, 22, 24, 26, 28, 31]).”
Old and New Examples of Scale Functions for Spectrally Negative Levy Processes (2007, background): “ory of queuing and storage models [7] and [25], in the theory of branching processes [11] and [45], in the theory of insurance risk and ruin [18, 20, 32, 33, 40, 42, 46, 52], in the theory of credit risk [29] and [43] and in the theory of fragmentation [37].”
Exit identities for L\'evy processes observed at Poisson arrival times (2014, methodology): “This approach builds upon some general formulas, see, for example, [12], Theorem 4, ignoring the crucial assumption of one-sided jumps.”
A note on scale functions and the time value of ruin for Levy insurance risk processes (2010, unknown): “Here one also finds a growing body of work (e.g.: Doney and Kyprianou, 2006; Kluppelberg and Erder, 2008; Breuer, 2008; Kyprianou, Pardo and Rivero, 2008a; Chaumont, Kyprianou and Pardo, 2009; Caballero and Chaumont, 2006; Chen and Sheu, 2009).”
Stopped diffusion processes: Boundary corrections and overshoot (2007, methodology): “Concerning the asymptotics of the overshoot of stochastic processes, let us mention the works of Alsmeyer [Als94] or Fuh and Lai [FL01] for ergodic Markov chains and Doney and Kyprianou for Lvy processes [DK06].”
Some explicit identities associated with positive self-similar Markov processes. (2007, background): “we have from Example 8 of Doney and Kyprianou [ 16 ] that the required probability is proportional to”
Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process (2007, background): “Email address: zpalma@math.uni.wroc.pl 428 et al. (2004), Doney and Kyprianou (2006), Klüppelberg and Kyprianou (2006), and Furrer (1998) in the context of insurance risk models.”
Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models (2010, methodology): “Therefore, a change of variables from t to x implies that the joint defective density of the surplus prior to ruin (x) and the deficit at ruin (y) for ruin occurring on the first claim is given by h∗1 (x, y |u ) = 1 c k ( x − u c ) p x−u c (x + y) , (8) and in this case the time of ruin T is (x −…”