Liang M, Wu B. Small deviations for admixture additive & multiplicative processes.
J Inequal Appl 2018;
2018:204. [PMID:
30839559 PMCID:
PMC6096921 DOI:
10.1186/s13660-018-1798-4]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/17/2018] [Accepted: 07/31/2018] [Indexed: 06/09/2023]
Abstract
Define the admixture additive processes X γ , H , α a 1 , a 2 , a 3 , a 4 ( t ) ≜ a 1 B ( t 1 ) + a 2 W γ ( t 2 ) + a 3 B H ( t 3 ) + a 4 S α ( t 4 ) ∈ R , and the admixture multiplicative processes Y γ , H , α ( t ) ≜ B ( t 1 ) ⋅ W γ ( t 2 ) ⋅ B H ( t 3 ) ⋅ S α ( t 4 ) ∈ R , where t = ( t 1 , t 2 , t 3 , t 4 ) ∈ R + 4 , a 1 , a 2 , a 3 , a 4 are finite constants, B ( t 1 ) is the standard Brownian motion, W γ ( t 2 ) is the fractional integrated Brownian motion with index parameter γ > - 1 / 2 , B H ( t 3 ) is the fractional Brownian motion with Hurst parameter H ∈ ( 0 , 1 ) , S α ( t 4 ) is the stable process with index α ∈ ( 0 , 2 ] , and they are independent of each other. The small deviation for X γ , H , α a 1 , a 2 , a 3 , a 4 ( t ) and the lower bound of small deviation for Y γ , H , α ( t ) are obtained. As an application, limit inf type LIL is given for X γ , H , α a 1 , a 2 , a 3 , a 4 ( t ) .
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