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36
Some remarks on first passage of Lévy processes, the American put and pasting principles
 Annals of Appl. Probability
"... 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 Lévy process and the solution of Gerber and Shiu [Astin ..."
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Cited by 31 (3 self)
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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 Lévy process and the solution of Gerber and Shiu [Astin
Smoothness of scale functions for spectrally negative Lévy processes
, 2006
"... Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that stan ..."
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Cited by 25 (8 self)
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Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.
DISTRIBUTIONAL STUDY OF DE FINETTI’S DIVIDEND PROBLEM FOR A GENERAL LÉVY Insurance Risk Process
 APPLIED PROBABILITY TRUST (15 MAY 2007)
, 2007
"... We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muller (2005) and Avram et al. (2006) which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically we build on recent wo ..."
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Cited by 21 (10 self)
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We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muller (2005) and Avram et al. (2006) which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically we build on recent work in the actuarial literature concerning calculations for the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than existing literature in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical CramérLundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integrodifferential equations and for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.
The law of the supremum of a stable Lévy process with no negative jumps
, 2006
"... Let X = (Xt)t≥0 be a stable Lévy process of index α ∈ (1, 2) with no negative jumps, and let St = sup 0≤s≤t Xs denote its running supremum for t>0. We show that the density function ft of St can be characterized as the unique solution to ..."
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Cited by 10 (2 self)
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Let X = (Xt)t≥0 be a stable Lévy process of index α ∈ (1, 2) with no negative jumps, and let St = sup 0≤s≤t Xs denote its running supremum for t>0. We show that the density function ft of St can be characterized as the unique solution to
The distribution of the maximum of a Lévy process with positive jumps of phasetype
 Proceedings of the Conference Dedicated to the 90th Anniversary of Boris Vladimirovich Gnedenko (Kyiv
, 2002
"... Consider a Levy process with nite intensity positive jumps of the phasetype and arbitrary negative jumps. Assume that the process either is killed at a constant rate or drifts to 1. We show that the distribution of the overall maximum of this process is also of phasetype, and nd the distribu ..."
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Cited by 8 (0 self)
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Consider a Levy process with nite intensity positive jumps of the phasetype and arbitrary negative jumps. Assume that the process either is killed at a constant rate or drifts to 1. We show that the distribution of the overall maximum of this process is also of phasetype, and nd the distribution of this random variable. Previous results (hyperexponential positive jumps) are obtained as a particular case.
Fluctuations of spectrally negative Markov additive processes
, 2008
"... For spectrally negative Markov Additive Processes (MAPs) we generalize classical fluctuation identities developed in Zolotarev (1964), Takács (1967), ..."
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Cited by 7 (1 self)
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For spectrally negative Markov Additive Processes (MAPs) we generalize classical fluctuation identities developed in Zolotarev (1964), Takács (1967),
American Options, Multi–Armed Bandits, and Optimal Consumption Plans: A Unified View
 IN PARIS–PRINCETON LECTURES IN FINANCIAL MATHEMATICS
"... In this survey, we show that various stochastic optimization problems arising in option theory, in dynamical allocation problems, and in the microeconomic theory of intertemporal consumption choice can all be reduced to the same problem of representing a given stochastic process in terms of running ..."
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Cited by 7 (2 self)
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In this survey, we show that various stochastic optimization problems arising in option theory, in dynamical allocation problems, and in the microeconomic theory of intertemporal consumption choice can all be reduced to the same problem of representing a given stochastic process in terms of running maxima of another process. We describe recent results of Bank and El Karoui (2002) on the general stochastic representation problem, derive results in closed form for Lévy processes and diffusions, present an algorithm for explicit computations, and discuss some applications.
Exotic Options under Lévy Models: An Overview
, 2004
"... In this paper we overview the pricing of several socalled exotic options in the nowdays quite popular exponential Lévy models. ..."
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Cited by 6 (0 self)
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In this paper we overview the pricing of several socalled exotic options in the nowdays quite popular exponential Lévy models.
Extremes of Markovadditive processes with onesided jumps, with queueing applications
, 2007
"... Through Laplace transforms, we study the extremes of a continuoustime Markovadditive process with onesided jumps and a finite background Markovian statespace, jointly with the epoch at which the extreme is ‘attained’. For this, we investigate discretetime Markovadditive processes and use an em ..."
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Cited by 5 (3 self)
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Through Laplace transforms, we study the extremes of a continuoustime Markovadditive process with onesided jumps and a finite background Markovian statespace, jointly with the epoch at which the extreme is ‘attained’. For this, we investigate discretetime Markovadditive processes and use an embedding to relate these to a continuoustime setting. The resulting Laplace transform is given in terms of two matrices, which can be determined either through solving a nonlinear matrix equation or through a spectral method. Our results on extremes are first applied to determine the steadystate buffercontent distribution of several singlestation queueing systems. We show that our framework comprises many models dealt with earlier, but, importantly, it also enables us to derive various new results. At the same time, our setup offers interesting insights into the connections between the approaches developed so far, including matrixanalytic techniques, martingale methods, the rateconservation approach, and the occupationmeasure method. Then we turn to tandem fluid networks, and show how the results on single queues can be used to find the Laplace transform of the steadystate buffercontent vector; it has a matrix quasiproduct
Limit theorems for stationary Markov processes with L 2 spectral
, 2012
"... Let (Xt,Yt)t∈T be a discrete or continuoustime Markov process with state space X ×R d where X is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. (Xt,Yt)t∈T is assumed to be a Markov additive process. In particular, this impl ..."
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Cited by 5 (4 self)
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Let (Xt,Yt)t∈T be a discrete or continuoustime Markov process with state space X ×R d where X is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. (Xt,Yt)t∈T is assumed to be a Markov additive process. In particular, this implies that the first component (Xt)t∈T is also a Markov process. Markovrandom walks or additive functionals of a Markovprocess are special instances of Markov additive processes. In this paper, the process (Yt)t∈T is shown to satisfy the following classical limit theorems: (a) the central limit theorem, (b) the local limit theorem, (c) the onedimensional BerryEsseen theorem, (d) the onedimensional firstorder Edgeworth expansion, provided that we have sup t∈(0,1]∩TEπ,0[Yt  α] < ∞ with the expected order α with respect to the independent case (up to some ε> 0 for (c) and (d)). For the statements (b) and (d), a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process (Xt)t∈T has an invariant probability distribution π, is stationary and has the L 2 (π)spectral gap property (that is, (Xt)t∈N is ρmixing in the discretetime case). The case where (Xt)t∈T is nonstationary is briefly discussed. As an application, we derive a BerryEsseen bound for the Mestimators associated with ρmixing Markov chains.