Asymptotic behavior of weighted cubic variation of sub-fractional brownian motion

ABSTRACT

In this article, we investigate the convergence of renormalized weighted cubic variation of a sub-fractional Brownian motion S^H with Hurst index H. When , we prove by means of Malliavin calculus that the convergence holds in L² toward an explicit limit which only depends on S^H. We also numerically simulate the sample paths of such a type of sub-fractional Brownian motion regulated by different Hurst index H.

Keywords:

• Malliavin calculus
• Sub-fractional Brownian motion
• Weighted cubic variation

Mathematics Subject Classification:

• 60H07
• 60G15
• 60G22

Acknowledgments

The authors would like to thank two reviewers for their careful reading and comments. Those comments and suggestions are valuable and very helpful for improving the manuscript.

Funding

Nenghui Kuang is partially supported by the National Natural Science Foundation of China under Grant 11101137 and by the Natural Science Foundation of Hunan Province under Grant 2015JJ2055 and by the Education Department Foundation of Hunan Province under Grant 14C0456. Huantian Xie is partially supported by the National Natural Science Foundation of China under Grants 11201212 and 11301252, the Natural Science Foundation of Shandong Province under Grant ZR2013FL006 and AMEP of Linyi University.