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Continuous-time methods in the study of discretely sampled functionals of Lévy processes. I. The positive process case

Part of:Probabilistic methods, simulation and stochastic differential equationsStochastic processes

Published online by Cambridge University Press:  01 July 2016

Michael Schröder
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Abstract

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In this paper we develop a constructive approach to studying continuously and discretely sampled functionals of Lévy processes. Estimates for the rate of convergence of the discretely sampled functionals to the continuously sampled functionals are derived, reducing the study of the latter to that of the former. Laguerre reduction series for the discretely sampled functionals are developed, reducing their study to that of the moment generating function of the pertinent Lévy processes and to that of the moments of these processes in particular. The results are applied to questions of contingent claim valuation, such as the explicit valuation of Asian options, and illustrated in the case of generalized inverse Gaussian Lévy processes.

Keywords

Lévy processorthogonal polynomialconstructive method for functionals of Lévy processesLaguerre reduction seriesgeneralized inverse Gaussian Lévy process

MSC classification

Primary:60G51: Processes with independent increments; L'evy processes65C50: Other computational problems in probability
Secondary:33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)33C90: Applications

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 1 , March 2007, pp. 245 - 270
Copyright
Copyright © Applied Probability Trust 2007 

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