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Gaussian Moving Averages and Semimartingales
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Andreas Basse, Department of Mathematical Sciences, University of Aarhus
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Abstract
In the present paper we study moving averages (also known as
stochastic convolutions) driven by a Wiener process and with a deterministic kernel.
Necessary and sufficient conditions on the kernel are provided for the moving average
to be a semimartingale in its natural filtration. Our results are constructive - meaning that
they provide a simple method to obtain kernels for which the moving average is a
semimartingale or a Wiener process. Several examples are considered.
In the last part of the paper we study general Gaussian processes with stationary
increments. We provide necessary and sufficient conditions on spectral
measure for the process to be a semimartingale.
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Full text: PDF
Pages: 1140-1165
Published on: July 22, 2008
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Bibliography
- Adler, Robert J. An introduction to continuity, extrema, and related topics for general Gaussian processes. Institute of Mathematical Statistics Lecture Notes---Monograph Series, 12. Institute of Mathematical Statistics, Hayward, CA, 1990. x+160 pp. ISBN: 0-940600-17-X MR1088478 (92g:60053)
- Basse, Andreas. Representation of Gaussian semimartingales with application to the covarians function. Thiele Centre -- Research Report 2008--05 (2008).
- Basse, Andreas. Spectral representation of Gaussian semimartingales. Thiele Centre -- Research Report 2008--03 (2008).
- Beurling, Arne. On two problems concerning linear transformations in Hilbert space. Acta Math. 81, (1948). 17 pp. MR0027954 (10,381e)
- Chaleyat-Maurel, Mireille; Jeulin, Thierry. Grossissement gaussien de la filtration brownienne. (French) [Gaussian enlargement of the Brownian filtration] C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 15, 699--702. MR0705695 (85a:60053)
- Cheridito, Patrick. Gaussian moving averages, semimartingales and option pricing. Stochastic Process. Appl. 109 (2004), no. 1, 47--68. MR2024843 (2004i:60049)
- Cherny, Alexander S. When is a moving average a semimartingal? MaPhySto -- Research Report 2001--28 (2001).
- Cramér, Harald. On some classes of nonstationary stochastic processes. 1961 Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II pp. 57--78 Univ. California Press, Berkeley, Calif. MR0150828 (27 #815)
- Doob, J. L.. The Brownian movement and stochastic equations. Ann. of Math. 2 (1942), no. 43, 351--369.
- Doob, J. L. Stochastic processes. Reprint of the 1953 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1990. viii+654 pp. ISBN: 0-471-52369-0 MR1038526 (91d:60002)
- Dym, H.; McKean, H. P. Gaussian processes, function theory, and the inverse spectral problem. Probability and Mathematical Statistics, Vol. 31. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. xi+335 pp. MR0448523 (56 #6829)
- Hardy, G. H.; Littlewood, J. E. Some properties of fractional integrals. I. Math. Z. 27 (1928), no. 1, 565--606. MR1544927
- Hida, Takeyuki. Canonical representations of Gaussian processes and their applications. Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math. 33 1960/1961 109--155. MR0119246 (22 #10012)
- Jeulin, T.; Yor, M. Filtration des ponts browniens et équations différentielles stochastiques linéaires. (French) [Filtration of Brownian bridges and linear stochastic differential equations] Séminaire de Probabilités, XXIV, 1988/89, 227--265, Lecture Notes in Math., 1426, Springer, Berlin, 1990. MR1071543 (91i:60206)
- Jeulin, T.; Yor, M. Moyennes mobiles et semimartingales. (French) [Moving averages and semimartingales] Séminaire de Probabilités, XXVII, 53--77, Lecture Notes in Math., 1557, Springer, Berlin, 1993. MR1308553 (95m:60119)
- Karhunen, Kari. Über die Struktur stationärer zufälliger Funktionen. (German) Ark. Mat. 1, (1950). 141--160. MR0034557 (11,607a)
- Knight, Frank B. Foundations of the prediction process. Oxford Studies in Probability, 1. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992. xii+248 pp. ISBN: 0-19-853593-7 MR1168699 (94b:60047)
- Lévy, Paul. Sur une classe de courbes de l'espace de Hilbert et sur une équation intégrale non linéaire. (French) Ann. Sci. Ecole Norm. Sup. (3) 73 1956 121--156. MR0096303 (20 #2787)
- Masani, P. On helixes in Hilbert space. I. Teor. Verojatnost. i Primenen. 17 (1972), 3--20. MR0298468 (45 #7520)
- Protter, Philip E. Stochastic integration and differential equations. Second edition. Applications of Mathematics (New York), 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004. xiv+415 pp. ISBN: 3-540-00313-4 MR2020294 (2005k:60008)
- Reinov, O. I. Functions of the first Baire class with values in metric spaces, and some of their applications. (Russian) Investigations on linear operators and the theory of functions, XIII. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 135 (1984), 135--149. MR0741703 (85f:46078)
- Rogers, L. C. G. Arbitrage with fractional Brownian motion. Math. Finance 7 (1997), no. 1, 95--105. MR1434408 (98b:90014)
- Stegall, Charles. Functions of the first Baire class with values in Banach spaces. Proc. Amer. Math. Soc. 111 (1991), no. 4, 981--991. MR1019283 (91k:26003)
- Stricker, C. Semimartingales gaussiennes---application au problème de l'innovation. (French) [Gaussian semimartingales---application to the innovation problem] Z. Wahrsch. Verw. Gebiete 64 (1983), no. 3, 303--312. MR0716488 (85c:60054)
- Yaglom, A. M. Correlation theory of stationary and related random functions. Vol. I. Basic results. Springer Series in Statistics. Springer-Verlag, New York, 1987. xiv+526 pp. ISBN: 0-387-96268-9 MR0893393 (89a:60105)
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Electronic Journal of Probability. ISSN: 1083-6489 |
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