Gaussian Volterra processes with power-type kernels. Part I
The stochastic process of the form \[ {X_{t}}={\int _{0}^{t}}{s^{\alpha }}\left({\int _{s}^{t}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\right)\hspace{0.1667em}d{W_{s}}\] is considered, where W is a standard Wiener process, $\alpha >-\frac{1}{2}$, $\gamma >-1$, and $\alpha +\beta +...
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| Format: | Article |
| Language: | English |
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VTeX
2022-04-01
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| Series: | Modern Stochastics: Theory and Applications |
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| Online Access: | https://www.vmsta.org/doi/10.15559/22-VMSTA205 |
| _version_ | 1818163482415071232 |
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| author | Yuliya Mishura Sergiy Shklyar |
| author_facet | Yuliya Mishura Sergiy Shklyar |
| author_sort | Yuliya Mishura |
| collection | DOAJ |
| description | The stochastic process of the form
\[ {X_{t}}={\int _{0}^{t}}{s^{\alpha }}\left({\int _{s}^{t}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\right)\hspace{0.1667em}d{W_{s}}\]
is considered, where W is a standard Wiener process, $\alpha >-\frac{1}{2}$, $\gamma >-1$, and $\alpha +\beta +\gamma >-\frac{3}{2}$. It is proved that the process X is well-defined and continuous. The asymptotic properties of the variances and bounds for the variances of the increments of the process X are studied. It is also proved that the process X satisfies the single-point Hölder condition up to order $\alpha +\beta +\gamma +\frac{3}{2}$ at point 0, the “interval” Hölder condition up to order $\min \big(\gamma +\frac{3}{2},\hspace{0.2222em}1\big)$ on the interval $[{t_{0}},T]$ (where $0<{t_{0}}<T$), and the Hölder condition up to order $\min \big(\alpha +\beta +\gamma +\frac{3}{2},\hspace{0.2778em}\gamma +\frac{3}{2},\hspace{0.2778em}1\big)$ on the entire interval $[0,T]$. |
| first_indexed | 2024-12-11T16:50:16Z |
| format | Article |
| id | doaj.art-c010189cfe224f9385ec54d8e3f54a30 |
| institution | Directory Open Access Journal |
| issn | 2351-6046 2351-6054 |
| language | English |
| last_indexed | 2024-12-11T16:50:16Z |
| publishDate | 2022-04-01 |
| publisher | VTeX |
| record_format | Article |
| series | Modern Stochastics: Theory and Applications |
| spelling | doaj.art-c010189cfe224f9385ec54d8e3f54a302022-12-22T00:58:07ZengVTeXModern Stochastics: Theory and Applications2351-60462351-60542022-04-019331333810.15559/22-VMSTA205Gaussian Volterra processes with power-type kernels. Part IYuliya Mishura0Sergiy Shklyar1Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64, Volodymyrska St., 01601 Kyiv, UkraineDepartment of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64, Volodymyrska St., 01601 Kyiv, UkraineThe stochastic process of the form \[ {X_{t}}={\int _{0}^{t}}{s^{\alpha }}\left({\int _{s}^{t}}{u^{\beta }}{(u-s)^{\gamma }}\hspace{0.1667em}du\right)\hspace{0.1667em}d{W_{s}}\] is considered, where W is a standard Wiener process, $\alpha >-\frac{1}{2}$, $\gamma >-1$, and $\alpha +\beta +\gamma >-\frac{3}{2}$. It is proved that the process X is well-defined and continuous. The asymptotic properties of the variances and bounds for the variances of the increments of the process X are studied. It is also proved that the process X satisfies the single-point Hölder condition up to order $\alpha +\beta +\gamma +\frac{3}{2}$ at point 0, the “interval” Hölder condition up to order $\min \big(\gamma +\frac{3}{2},\hspace{0.2222em}1\big)$ on the interval $[{t_{0}},T]$ (where $0<{t_{0}}<T$), and the Hölder condition up to order $\min \big(\alpha +\beta +\gamma +\frac{3}{2},\hspace{0.2778em}\gamma +\frac{3}{2},\hspace{0.2778em}1\big)$ on the entire interval $[0,T]$.https://www.vmsta.org/doi/10.15559/22-VMSTA205Gaussian Volterra processesfractional Brownian motionHölder continuityquasi-helix property |
| spellingShingle | Yuliya Mishura Sergiy Shklyar Gaussian Volterra processes with power-type kernels. Part I Modern Stochastics: Theory and Applications Gaussian Volterra processes fractional Brownian motion Hölder continuity quasi-helix property |
| title | Gaussian Volterra processes with power-type kernels. Part I |
| title_full | Gaussian Volterra processes with power-type kernels. Part I |
| title_fullStr | Gaussian Volterra processes with power-type kernels. Part I |
| title_full_unstemmed | Gaussian Volterra processes with power-type kernels. Part I |
| title_short | Gaussian Volterra processes with power-type kernels. Part I |
| title_sort | gaussian volterra processes with power type kernels part i |
| topic | Gaussian Volterra processes fractional Brownian motion Hölder continuity quasi-helix property |
| url | https://www.vmsta.org/doi/10.15559/22-VMSTA205 |
| work_keys_str_mv | AT yuliyamishura gaussianvolterraprocesseswithpowertypekernelsparti AT sergiyshklyar gaussianvolterraprocesseswithpowertypekernelsparti |