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18
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 78 (18 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.
Lévydriven polling systems and continuousstate branching processes
, 2009
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The distribution of the maximal difference between Brownian bridge and its concave majorant
, 2009
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Supplement to “Power Algorithms for Inverting Laplace Transforms”
"... This is a supplement to the main paper, presenting additional experimental results not included in the main paper due to lack of space. The overall study investigates ways to create algorithms to numerically invert Laplace transforms within a unified framework proposed by Abate and Whitt (2006). Tha ..."
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Cited by 1 (0 self)
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This is a supplement to the main paper, presenting additional experimental results not included in the main paper due to lack of space. The overall study investigates ways to create algorithms to numerically invert Laplace transforms within a unified framework proposed by Abate and Whitt (2006). That framework approximates the desired function value by a finite linear combination of transform values, depending on parameters called weights and nodes, which are initially left unspecified. Alternative parameter sets, and thus algorithms, are generated and evaluated here by considering power test functions. The resulting power algorithms are shown to be effective, with the parameter choice being tunable to the transform being inverted. The powers can be advantageously chosen from series expansions of the transform. Real weights for a realvariable power algorithm are found for specified real powers and positive real nodes by solving a system of linear equations involving a generalized Vandermonde matrix, using Mathematica. Experiments show that the power algorithms are robust in the nodes; it suffices to use the first n positive integers. The power test functions also provide a useful way to evaluate the performance of other algorithms. History: draft aiming for JoC. Last modified on December 9, 2006.
Advection, diffusion, and delivery over a network, Phys
 Rev. E 86 (2012) 021905. doi:10.1103/PhysRevE.86.021905. URL http://arxiv.org/abs/1105.1647
"... Many biological, geophysical and technological systems involve the transport of resource over a network. In this paper we present an algorithm for calculating the exact concentration of resource at any point in space or time, given that the resource in the network is lost or delivered out of the ne ..."
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Many biological, geophysical and technological systems involve the transport of resource over a network. In this paper we present an algorithm for calculating the exact concentration of resource at any point in space or time, given that the resource in the network is lost or delivered out of the network at a given rate, while being subject to advection and diffusion. We consider the implications of advection, diffusion and delivery for simple models of glucose delivery through a vascular network, and conclude that in certain circumstances, increasing the volume of blood and the number of glucose transporters can actually decrease the total rate of glucose delivery. We also consider the case of empirically determined fungal networks, and analyze the distribution of resource that emerges as such networks grow over time. Fungal growth involves the expansion of fluid filled vessels, which necessarily involves the movement of fluid. In three empirically determined fungal networks we found that the minimum currents consistent with the observed growth would effectively transport resource throughout the network over the timescale of growth. This suggests that in foraging fungi, the active transport mechanisms observed in the growing tips may not be required for long range transport.
Exact power spectra of Brownian motion with solid friction
"... Abstract. We study a Langevin equation describing the Brownian motion of an object subjected to a viscous drag, an external constant force, and a solid friction force of the Coulomb type. In a previous work [H. Touchette, E. Van der Straeten, W. Just, J. Phys. A: Math. Theor. 43, 445002, 2010], we h ..."
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Abstract. We study a Langevin equation describing the Brownian motion of an object subjected to a viscous drag, an external constant force, and a solid friction force of the Coulomb type. In a previous work [H. Touchette, E. Van der Straeten, W. Just, J. Phys. A: Math. Theor. 43, 445002, 2010], we have presented the exact solution of the velocity propagator of this equation based on a spectral decomposition of the corresponding FokkerPlanck equation. Here, we present an alternative, exact solution based on the Laplace transform of this equation, which has the advantage of being expressed in closed form. From this solution, we also obtain closedform expressions for the Laplace transform of the velocity autocorrelation function and for the power spectrum, i.e., the Fourier transform of the autocorrelation function. The behavior of the power spectrum as a function of the dry friction force and external forcing shows a clear crossover between stick and slip regimes known to occur in the presence of solid friction. PACS numbers: 05.40.Jc, 46.55.+d, 05.10.Gg, 02.30.Jr, 02.50.Cw 1.
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"... Radial transport in a porous medium with Dirichlet, Neumann and Robintype inhomogeneous boundary values and general initial data: analytical solution and evaluation ..."
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Radial transport in a porous medium with Dirichlet, Neumann and Robintype inhomogeneous boundary values and general initial data: analytical solution and evaluation
E Receptors: a Dialog between Experiment and Theory
, 2012
"... This dissertation is approved, and it is acceptable in quality and form for publication: Approved by the Dissertation Committee: ..."
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This dissertation is approved, and it is acceptable in quality and form for publication: Approved by the Dissertation Committee:
Pricing and hedging barrier options driven by a class
, 812
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