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A Lévy process on Rd with distribution }μ at time 1 is denoted by Xμ = {Xμt}, If the improper stochastic integral [formula] of f with respect to Xμ is definable, its distribution is denoted by Ф∫(μ). The class of all infinitely divisible distributions μ on Rd such that Ф∫(μ) is definable is denoted by D(Φ∫). The class D(Φ∫), its two extensions Dc(Φ∫) and Des(Φ∫) (compensated and essential), and its restriction D0(Φ∫)(absolutely definable) are studied. It is shown that Des(Φ∫) is monotonic with respect to ∫, which means that |f2| ≤ |f1| implies Des(Φ∫1) ⊂ Des(Φ∫2). Further D0Φf is monotonic with respect to ∫ but neither D(Φ∫) nor D0(Φ∫)is monotonic with respect to ∫. Furthermore, there exist μ, ∫1 and ∫2 such that 0 ≤∫2 ∫1, and μ∈D(Φ∫1), and μ ∉D(Φ∫2) An explicit example for this is related to some properties of a class of martingale Levy processes.
Czasopismo
Rocznik
Tom
Strony
23--39
Opis fizyczny
Bibliogr. 16 poz.
Twórcy
Bibliografia
- [1] O.E. Barndorff-Nielsen, M. Maejima and K. Sato, Some classes of infinitely divisible distributions admitting stochastic integral representations, Bernoulli 12 (2006), pp. 1-33.
- [2] A. Cherny and A. Shiryaev, On stochastic integrals up to infinity and predictable criteria for integrability, Lecture Notes in Math. No 1857, Springer, 2005, pp. 165-185.
- [3] Z.J. Jurek, The class L,(Q) of probability measures on Banach spaces, Bull. Polish Acad. Sci. Math. 31 (1983), pp. 51-62.
- [4] Z. J. Jurek and W. Vervaat, An integral representation far selfdecomposable Banach space valued random variables, 2. Wahrsch. Verw. Gebietc 62 (1983), pp. 247-262.
- [5] S. Kwapień and W. A. Woyczyński, Random Series and Stochastic Integrals. Single and Multiple, Birkhäuser, Boston 1992.
- [6] B. Rajput and J. Rosinski, Spectral representations of linfinitely divisible processes, Probab. Theory Related Fields 82 (1989), pp. 451-487.
- [7] K. Sato, Class L of multivariate distributions and its subclasses, J. Multivariate Anal. 10 (1980), pp. 207-232.
- [8] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge Univ. Press, Cambridge1999.
- [9] K. Sato, Stochastic integrals in additive processes and application to semi-Lévy processes, Osaka J. Math. 41 (20041, pp. 211-236.
- [10] K. Sato, Additive processes and stochastic integrals, Illinois J. Math. 50 (2006) (Doob Volume), pp. 825-851.
- [11] K. Sato, Two families of improper stochastic integrals with respect to Lévy processes, Alea, Latin American Journal of Probability and Mathematical Statistics 1 (2006), pp. 47-87. http://alea.impa.br/english/
- [12] K. Urbanik, Slowly varying sequences of random variables, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys, 20 (1972), pp. 679-682.
- [13] K. Urbanik, Limit laws for sequences of normed sums satisfying some stability conditions, in: Multivariate Analysis - III, P. R. Krishnaiah (Ed.), Academic Press, New York, 1973, pp. 225-237.
- [14] K. Urbanik and W, A Woyczynski, Random integrals and Orlicz spaces, Bull. Acad. Polon. Sci., Sir. Sci. Math. Astronom. Phys. 15 (1967), pp. 161-169.
- [I5] S. J. Wolfe, A characterization of certain stochastic integrals in: Tenth Conference on Stochastic Processes and Their Applications. Contributed Papers, Stochastic Process. Appl. 12 (1982), p. 136.
- [l6] S. J. Wolfe, On a continuous analogue of the stochastic difference equation X,= qX,- +B,, Stochastic Process. Appl. 12 (19821, pp. 301-312.
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Bibliografia
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