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** method uses the analytic continuation of the Lévy density to a complex cone and contour integration. As a by-product, we investigate the smoothness of the semigroup density in time ** – use for probability equation on macro-econ model – US conditional approach / sequencing / by flow rates:time / other factors
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http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bj/1219669629

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Bernoulli

Probability measures, Lévy measures and analyticity in time

Ole E. Barndorff-Nielsen and Friedrich Hubalek

Source: Bernoulli Volume 14, Number 3 (2008), 764-790.
Abstract

We investigate the relation of the semigroup probability density of an infinite activity Lévy process to the corresponding Lévy density. For subordinators, we provide three methods to compute the former from the latter. The first method is based on approximating compound Poisson distributions, the second method uses convolution integrals of the upper tail integral of the Lévy measure and the third method uses the analytic continuation of the Lévy density to a complex cone and contour integration. As a by-product, we investigate the smoothness of the semigroup density in time. Several concrete examples illustrate the three methods and our results.
Keywords: cancellation of singularities; exponential formula; generalised gamma convolutions; subordinators

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Permanent link to this document: http://projecteuclid.org/euclid.bj/1219669629
Digital Object Identifier: doi:10.3150/07-BEJ6114
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References
Abramowitz, M. and Stegun, I.A. eds. (1992). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications Inc.
Barndorff-Nielsen, O.E. (2000). Probability densities and Lévy densities. Research Report 18, Aarhus Univ., Centre for Mathematical Physics and Stochastics (MaPhySto).
Barndorff-Nielsen, O.E. and Blæsild, P. (1983). Reproductive exponential families. Ann. Statist. 11 770–782.
Barndorff-Nielsen, O.E. and Hubalek, F. (2006). Probability measures, Lévy measures, and analyticity in time. Thiele Research Report 2006–12, Univ. Aarhus, Denmark, The Thiele Centre.
Barndorff-Nielsen, O.E. and Shephard, N. (2001). Modelling by Lévy processes for financial econometrics. In Lévy processes, Theory and Applications (O.E. Barndorff-Nielsen, T. Mikosch and S.I. Resnick, eds.) 283–318. Boston, MA: Birkhäuser.
Barndorff-Nielsen, O.E. and Shephard, N. (2008). Financial Volatility in Continuous Time: Volatility and Lévy Based Modelling. Cambridge Univ. Press. To appear.
Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions and Densities. New York: Springer.
Burnaev, E.V. (2006). An inversion formula for infinitely divisible distributions. Uspekhi Matematicheskikh Nauk 61 187–188.
Comtet, L. (1970). Analyse combinatoire. Tome I. Paris: Presses Universitaires de France.
Doetsch, G. (1950). Handbuch der Laplace-Transformation. 1. Basel: Birkhäuser.
Doney, R.A. (2004). Small-time behaviour of Lévy processes. Electron. J. Probab. 9 209–229.
Embrechts, P. and Goldie, C.M. (1981). Comparing the tail of an infinitely divisible distribution with integrals of its Lévy measure. Ann. Probab. 9 468–481.
Embrechts, P., Goldie, C.M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrsch. Verw. Gebiete 49 335–347.
Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. (1954). Tables of Integral Transforms. I. New York: McGraw-Hill. (15,868a)
Feller, W. (1971). An Introduction to Probability Theory and Its Applications. II, 2nd ed. New York: Wiley. (42 #5292)
Gradshteyn, I.S. and Ryzhik, I.M. (2000). Table of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press Inc.
Hille, E. and Phillips, R.S. (1957). Functional Analysis and Semi-groups, rev. ed. Providence, RI: Amer. Math. Soc.
Hubalek, F. (2002). On a conjecture of Barndorff-Nielsen relating probability densities and Lévy densities. In Proceedings of the 2nd MaPhySto Conference on Lévy Processes: Theory and Applications (O.E. Barndorff-Nielsen, ed.). Number 22 in MaPhySto Miscellanea.
Ishikawa, Y. (1994). Asymptotic behavior of the transition density for jump type processes in small time. Tohoku Math. J. Second Series 46 443–456.
Kawata, T. (1972). Fourier Analysis in Probability Theory. New York: Academic Press.
Léandre, R. (1987). Densité en temps petit d’un processus de sauts. Séminaire de Probabilités XXI. Lecture Notes in Math. 1247 81–99. Berlin: Springer. (89g:60179)
Picard, J. (1997). Density in small time at accessible points for jump processes. Stochastic Process. Appl. 67 251–279.
Rüschendorf, L. and Woerner, J.H.C. (2002). Expansion of transition distributions of Lévy processes in small time. Bernoulli 8 81–96.
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge Univ. Press.
Sato, K. and Steutel, F.W. (1998). Note on the continuation of infinitely divisible distributions and canonical measures. Statistics 31 347–357.
Schoutens, W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives. Chichester: Wiley.
Uludag(, M. (1998). On possible deterioration of smoothness under the operation of convolution. J. Math. Anal. Appl. 227 335–358.
Woerner, J.H.C. (2001). Statistical analysis for discretely observed Lévy processes. Dissertation, Freiburg Univ.
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©2008 Cornell University Library

Barndorff-Nielsen, O.E. and Shephard, N. (2008). Financial Volatility in Continuous Time: Volatility and Lévy Based Modelling. Cambridge Univ. Press. To appear.

Burnaev, E.V. (2006). An inversion formula for infinitely divisible distributions. Uspekhi Matematicheskikh Nauk 61 187–188.

Barndorff-Nielsen, O.E. and Hubalek, F. (2006). Probability measures, Lévy measures, and analyticity in time. Thiele Research Report 2006–12, Univ. Aarhus, Denmark, The Thiele Centre.

Uludag(, M. (1998). On possible deterioration of smoothness under the operation of convolution. J. Math. Anal. Appl. 227 335–358.

2008 © Bernoulli Society for Mathematical Statistics and Probability

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bj/1219669629

** third method uses the analytic continuation of the Lévy density to a complex cone and contour integration. **

Probability measures, Lévy measures and analyticity in time

Ole E. Barndorff-Nielsen and Friedrich Hubalek

Source: Bernoulli Volume 14, Number 3 (2008), 764-790.
Abstract

We investigate the relation of the semigroup probability density of an infinite activity Lévy process to the corresponding Lévy density. For subordinators, we provide three methods to compute the former from the latter. The first method is based on approximating compound Poisson distributions, the second method uses convolution integrals of the upper tail integral of the Lévy measure and the third method uses the analytic continuation of the Lévy density to a complex cone and contour integration. As a by-product, we investigate the smoothness of the semigroup density in time. Several concrete examples illustrate the three methods and our results.
Keywords: cancellation of singularities; exponential formula; generalised gamma convolutions; subordinators

©2008 Cornell University Library

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