Time Regularity of Generalized Ornstein–Uhlenbeck Processes with Lévy Noises in Hilbert Spaces

Yong Liu () and Jianliang Zhai ()
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Yong Liu: Peking University
Jianliang Zhai: University of Science and Technology of China

Journal of Theoretical Probability, 2016, vol. 29, issue 3, 843-866

Abstract: Abstract In this paper we first obtain a necessary condition for $$H$$ H -càdlàg modification and $$H$$ H -weakly càdlàg modification of generalized Ornstein–Uhlenbeck processes with Lévy noises in Hilbert spaces $$H$$ H . Then we give a necessary and sufficient condition for the $$H$$ H -càdlàg modification and $$H$$ H -weakly càdlàg modification of Ornstein–Uhlenbeck processes driven by cylindrical $$\alpha $$ α -semistable processes. Secondly, we investigate the properties of cylindrical càdlàg modification and $$V$$ V -cylindrical càdlàg modification. Applying the obtained results to diagonal Ornstein–Uhlenbeck processes with $$\alpha $$ α -stable noises, we show a necessary and sufficient condition for cylindrical càdlàg modification and $$V$$ V -cylindrical càdlàg modification in the symmetric case for $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) and give a sufficient condition in the general case for $$\alpha \in (0,2)$$ α ∈ ( 0 , 2 ) . Some examples illustrate the relations among the concepts of various càdlàg modifications.

Keywords: Generalized Ornstein–Uhlenbeck processes with Lévy noises; Càdlàg modification; $$\alpha $$ α -stable process; $$\alpha $$ α -semistable process; 60H15; 60J75 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10959-015-0594-z

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