A law of iterated logarithm for the subfractional Brownian motion and an application
Abstract Let SH={StH,t≥0} $S^{H}=\{S^{H}_{t},t\geq0\}$ be a sub-fractional Brownian motion with Hurst index 0<H<1 $0< H<1$. In this paper, we give a local law of the iterated logarithm of the form lim sups↓0|St+sH−StH|sH2log+log(1/s)=1, $$\limsup_{s\downarrow0}\frac{ \vert S^{H}_{t+s}-S^...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-04-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-018-1675-1 |
| _version_ | 1818319484240265216 |
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| author | Hongsheng Qi Litan Yan |
| author_facet | Hongsheng Qi Litan Yan |
| author_sort | Hongsheng Qi |
| collection | DOAJ |
| description | Abstract Let SH={StH,t≥0} $S^{H}=\{S^{H}_{t},t\geq0\}$ be a sub-fractional Brownian motion with Hurst index 0<H<1 $0< H<1$. In this paper, we give a local law of the iterated logarithm of the form lim sups↓0|St+sH−StH|sH2log+log(1/s)=1, $$\limsup_{s\downarrow0}\frac{ \vert S^{H}_{t+s}-S^{H}_{t} \vert }{ s^{H}\sqrt {2\log^{+}\log(1/s)}}=1, $$ almost surely, for all t>0 $t > 0$, where log+x=max{1,logx} $\log^{+}x=\max{\{1, \log x\}}$ for x≥0 $x\geq0$. As an application, we introduce the ΦH $\Phi_{H}$-variation of SH $S^{H}$ driven by ΦH(x):=[x/2log+log+(1/x)]1/H $\Phi_{H}(x):= [x/\sqrt{2\log^{+}\log ^{+}(1/x)} ]^{1/H}$ (x>0) $(x>0)$ with ΦH(0)=0 $\Phi_{H}(0)=0$. |
| first_indexed | 2024-12-13T10:09:51Z |
| format | Article |
| id | doaj.art-c2c91a9d39cf41588a147a4504ed8d6a |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-13T10:09:51Z |
| publishDate | 2018-04-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-c2c91a9d39cf41588a147a4504ed8d6a2022-12-21T23:51:28ZengSpringerOpenJournal of Inequalities and Applications1029-242X2018-04-012018111810.1186/s13660-018-1675-1A law of iterated logarithm for the subfractional Brownian motion and an applicationHongsheng Qi0Litan Yan1Department of Mathematics, College of Science, Bengbu UniversityDepartment of Mathematics, College of Science, Donghua UniversityAbstract Let SH={StH,t≥0} $S^{H}=\{S^{H}_{t},t\geq0\}$ be a sub-fractional Brownian motion with Hurst index 0<H<1 $0< H<1$. In this paper, we give a local law of the iterated logarithm of the form lim sups↓0|St+sH−StH|sH2log+log(1/s)=1, $$\limsup_{s\downarrow0}\frac{ \vert S^{H}_{t+s}-S^{H}_{t} \vert }{ s^{H}\sqrt {2\log^{+}\log(1/s)}}=1, $$ almost surely, for all t>0 $t > 0$, where log+x=max{1,logx} $\log^{+}x=\max{\{1, \log x\}}$ for x≥0 $x\geq0$. As an application, we introduce the ΦH $\Phi_{H}$-variation of SH $S^{H}$ driven by ΦH(x):=[x/2log+log+(1/x)]1/H $\Phi_{H}(x):= [x/\sqrt{2\log^{+}\log ^{+}(1/x)} ]^{1/H}$ (x>0) $(x>0)$ with ΦH(0)=0 $\Phi_{H}(0)=0$.http://link.springer.com/article/10.1186/s13660-018-1675-1Sub-fractional Brownian motionIterated logarithmΦ-variation |
| spellingShingle | Hongsheng Qi Litan Yan A law of iterated logarithm for the subfractional Brownian motion and an application Journal of Inequalities and Applications Sub-fractional Brownian motion Iterated logarithm Φ-variation |
| title | A law of iterated logarithm for the subfractional Brownian motion and an application |
| title_full | A law of iterated logarithm for the subfractional Brownian motion and an application |
| title_fullStr | A law of iterated logarithm for the subfractional Brownian motion and an application |
| title_full_unstemmed | A law of iterated logarithm for the subfractional Brownian motion and an application |
| title_short | A law of iterated logarithm for the subfractional Brownian motion and an application |
| title_sort | law of iterated logarithm for the subfractional brownian motion and an application |
| topic | Sub-fractional Brownian motion Iterated logarithm Φ-variation |
| url | http://link.springer.com/article/10.1186/s13660-018-1675-1 |
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