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On stochastic integral representation of stable processes with sample paths in Banach spaces

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  • Rosinski, Jan

Abstract

Certain path properties of a symmetric [alpha]-stable process X(t) = [integral operator]Sh(t, s) dM(s), t [set membership, variant] T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double [alpha]-stable integral. Also, necessary and sufficient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to[13], Ann. Probab.9, 624-632) is extensively used and the relationship between these two representations is discussed.

Suggested Citation

  • Rosinski, Jan, 1986. "On stochastic integral representation of stable processes with sample paths in Banach spaces," Journal of Multivariate Analysis, Elsevier, vol. 20(2), pages 277-302, December.
    • Handle: RePEc:eee:jmvana:v:20:y:1986:i:2:p:277-302
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    Cited by:

    1. Yu. Davydov & V. Paulauskas & A. Račkauskas, 2000. "More on P-Stable Convex Sets in Banach Spaces," Journal of Theoretical Probability, Springer, vol. 13(1), pages 39-64, January.
    2. Dozzi, Marco & Soltani, A. Reza, 1997. "Local time for stable moving average processes: Hölder conditions," Stochastic Processes and their Applications, Elsevier, vol. 68(2), pages 195-207, June.
    3. Fabian A. Harang & Chengcheng Ling, 2022. "Regularity of Local Times Associated with Volterra–Lévy Processes and Path-Wise Regularization of Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1706-1735, September.

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