Multiparameter processes with stationary increments: Spectral representation and integration

Andreas Basse-O'Connor (Aarhus University)
Svend-Erik Graversen (Aarhus University)
Jan Pedersen (Aarhus University)

Abstract


In this article,  a class of multiparameter processes with wide-sense  stationary increments is studied. The content is as follows. (1) The spectral representation is derived;  in particular, necessary and sufficient conditions for a measure to be a spectral measure is given. The relations to a  commonly used  class  of processes, studied e.g.  by Yaglom, is discussed. (2) Some classes of deterministic integrands, here referred to as predomains,  are studied in detail. These predomains consist of functions or,  more generally, distributions. Necessary and sufficient conditions for completeness of the predomains are given. (3) In a framework covering the classical Walsh-Dalang theory of a temporal-spatial process which is white in time and colored in space, a class of predictable integrands is considered. Necessary and sufficient conditions for completeness of the class  are given, and this property is linked to a certain martingale representation property.

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Pages: 1-21

Publication Date: September 5, 2012

DOI: 10.1214/EJP.v17-2287

References

  • Bonami, Aline; Estrade, Anne. Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9 (2003), no. 3, 215--236. MR1988750
  • Dalang, Robert C. Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s. Electron. J. Probab. 4 (1999), no. 6, 29 pp. (electronic). MR1684157
  • J. Hoffmann-Jørgensen, Probability with a view toward statistics. Vol. I, Chapman & Hall Probability Series, Chapman & Hall, New York, 1994.MR1278485
  • Itô, Kiyosi. Stationary random distributions. Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math. 28, (1954). 209--223. MR0065060
  • Jolis, Maria. The Wiener integral with respect to second order processes with stationary increments. J. Math. Anal. Appl. 366 (2010), no. 2, 607--620. MR2600506
  • Kolmogoroff, A. Kurven im Hilbertschen Raum, die gegenüber einer einparametrigen Gruppe von Bewegungen invariant sind. (German) C. R. (Doklady) Acad. Sci. URSS (N.S.) 26, (1940). 6--9. MR0003440
  • Pipiras, Vladas; Taqqu, Murad S. Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 (2000), no. 2, 251--291. MR1790083
  • Rogers, L. C. G.; Williams, David. Diffusions, Markov processes, and martingales. Vol. 1. Foundations. Reprint of the second (1994) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. xx+386 pp. ISBN: 0-521-77594-9 MR1796539
  • Schwartz, Laurent. Théorie des distributions. (French) Publications de l'Institut de Mathématique de l'Université de Strasbourg, No. IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée. Hermann, Paris 1966 xiii+420 pp. MR0209834
  • Terdik, György; Woyczyński, Wojbor A. Notes on fractional Ornstein-Uhlenbeck random sheets. Publ. Math. Debrecen 66 (2005), no. 1-2, 153--181. MR2128690
  • von Neumann, J.; Schoenberg, I. J. Fourier integrals and metric geometry. Trans. Amer. Math. Soc. 50, (1941). 226--251. MR0004644
  • Walsh, John B. An introduction to stochastic partial differential equations. École d'été de probabilités de Saint-Flour, XIV—1984, 265--439, Lecture Notes in Math., 1180, Springer, Berlin, 1986. MR0876085
  • Yaglom, A. M. Certain types of random fields in $n$-dimensional space similar to stationary stochastic processes. (Russian) Teor. Veroyatnost. i Primenen 2 1957 292--338. MR0094844
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