Operator self-similar processes on Banach spaces

Mihaela T. Matache and Valentin Matache

International Journal of Stochastic Analysis, 2006, vol. 2006, 1-18

Abstract:

Operator self-similar (OSS) stochastic processes on arbitrary Banach spaces are considered. If the family of expectations of such a process is a spanning subset of the space, it is proved that the scaling family of operators of the process under consideration is a uniquely determined multiplicative group of operators. If the expectation-function of the process is continuous, it is proved that the expectations of the process have power-growth with exponent greater than or equal to 0 , that is, their norm is less than a nonnegative constant times such a power-function, provided that the linear space spanned by the expectations has category 2 (in the sense of Baire) in its closure. It is shown that OSS processes whose expectation-function is differentiable on an interval ( s 0 , ∞ ) , for some s 0 ≥ 1 , have a unique scaling family of operators of the form { s H : s > 0 } , if the expectations of the process span a dense linear subspace of category 2 . The existence of a scaling family of the form { s H : s > 0 } is proved for proper Hilbert space OSS processes with an Abelian scaling family of positive operators.

Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:082838

DOI: 10.1155/JAMSA/2006/82838

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