×

Random integral representation of operator-semi-self-similar processes with independent increments. (English) Zbl 1075.60029

Summary: M. Jeanblanc et al. give a representation of self-similar processes with independent increments by stochastic integrals with respect to background driving Lévy processes. Via Lamperti’s transformation these processes correspond to stationary Ornstein-Uhlenbeck processes. In the present paper we generalize the integral representation to multivariate processes with independent increments having the weaker scaling property of operator-semi-self-similarity. It turns out that the corresponding background driving process has periodically stationary increments and in general is no longer a Lévy process. Just as well it turns out that the Lamperti transform of an operator-semi-self-similar process with independent increments defines a periodically stationary process of Ornstein-Uhlenbeck type.

MSC:

60G18 Self-similar stochastic processes
60G51 Processes with independent increments; Lévy processes
60H05 Stochastic integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Becker-Kern, P., Stable and semistable hemigroupsdomains of attraction and selfdecomposability, J. Theoret. Probab., 16, 573-598 (2001)
[2] Fitzsimmons, P.; Pitman, J.; Yor, M., Markovian bridges: construction, Palm interpretation, and splicing, (Cinlar, E.et al., Seminar on Stochastic Processes, 1992 (1993), Birkhäuser: Birkhäuser Boston), 101-134 · Zbl 0844.60054
[3] Hazod, W.; Scheffler, H. P., Strongly \(τ\)-decomposable and selfdecomposable laws on simply connected nilpotent Lie groups, Monatsh. Math., 128, 269-282 (1999) · Zbl 0947.60005
[4] Hudson, W. N.; Mason, J. D., Operator-self-similar processes in a finite-dimensional space, Trans. Amer. Math. Soc., 273, 281-297 (1982) · Zbl 0508.60044
[5] Hurd, H. L., Stationarizing properties of random shifts, SIAM J. Appl. Math., 26, 203-212 (1974) · Zbl 0276.60043
[6] Jacod, J.; Shiryaev, A. N., Limit Theorems for Stochastic Processes (1987), Springer: Springer Berlin · Zbl 0635.60021
[7] Jeanblanc, M.; Pitman, J.; Yor, M., Self-similar processes with independent increments associated with Lévy and Bessel processes, Stochastic Process. Appl., 100, 223-231 (2002) · Zbl 1059.60052
[8] Jones, R. H.; Brelsford, W. M., Time series with periodic structure, Biometrika, 54, 403-408 (1967) · Zbl 0153.47706
[9] Jurek, Z. J., An integral representation of operator-selfdecomposable random variables, Bull. Acad. Polon. Sci., 30, 385-393 (1982) · Zbl 0503.60063
[10] Jurek, Z. J., Selfdecomposabilityan exception or a rule?, Ann. Univ. Mariae Curie-Skłodowska Sect. A, 51, 93-107 (1997) · Zbl 0904.60012
[11] Jurek, Z. J.; Mason, J. D., Operator-Limit Distributions in Probability Theory (1993), Wiley: Wiley New York · Zbl 0850.60003
[12] Jurek, Z. J.; Vervaat, W., An integral representation for self-decomposable Banach space valued random variables, Z. Wahrsch. Verw. Gebiete, 62, 247-262 (1983) · Zbl 0488.60028
[13] Lamperti, J., Semi-stable stochastic processes, Trans. Amer. Math. Soc., 104, 62-78 (1962) · Zbl 0286.60017
[14] Maejima, M.; Sato, K. I., Semi-selfsimilar processes, J. Theoret. Probab., 12, 347-373 (1999) · Zbl 0932.60043
[15] Maejima, M.; Sato, K. I.; Watanabe, T., Distributions of selfsimilar and semi-selfsimilar processes with independent increments, Statist. Probab. Lett., 47, 395-401 (2000) · Zbl 0955.60038
[16] Meerschaert, M. M.; Scheffler, H. P., Limit Distributions for Sums of Independent Random Vectors (2001), Wiley: Wiley New York · Zbl 0990.60003
[17] Pedersen, J., Periodic Ornstein-Uhlenbeck processes driven by Lévy processes, J. Appl. Probab., 39, 748-763 (2002) · Zbl 1023.60041
[18] Sato, K. I., Self-similar processes with independent increments, Probab. Theory Relat. Fields, 89, 285-300 (1991) · Zbl 0725.60034
[19] Sato, K. I., Lévy Processes and Infinitely Divisible Distributions (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0973.60001
[20] Sato, K. I.; Yamamuro, K., On selfsimilar and semi-selfsimilar processes with independent increments, J. Korean Math. Soc., 35, 207-224 (1998) · Zbl 0902.60032
[21] Wolfe, S. J., On a continuous analogue of the stochastic difference equation \(X_n=ρX_{n\)−1 · Zbl 0482.60062
[22] Wolfe, S. J., A characterization of certain stochastic integrals, Stochastic Process. Appl., 12, 136 (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.