The quadratic variation for mixed-fractional Brownian motion
Abstract Let W = λ B + ν B H ${W}=\lambda B+\nu B^{H}$ be a mixed-fractional Brownian motion with Hurst index 0 < H < 1 2 $0< H<\frac{1}{2}$ and λ , ν ≠ 0 $\lambda,\nu\neq0$ . In this paper we study the quadratic covariation [ f ( W ) , W ] ( H ) $[f({W}),{W}]^{(H)}$ defined by [ f ( W )...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2016-11-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | http://link.springer.com/article/10.1186/s13660-016-1254-2 |
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| author | Han Gao Kun He Litan Yan |
| author_facet | Han Gao Kun He Litan Yan |
| author_sort | Han Gao |
| collection | DOAJ |
| description | Abstract Let W = λ B + ν B H ${W}=\lambda B+\nu B^{H}$ be a mixed-fractional Brownian motion with Hurst index 0 < H < 1 2 $0< H<\frac{1}{2}$ and λ , ν ≠ 0 $\lambda,\nu\neq0$ . In this paper we study the quadratic covariation [ f ( W ) , W ] ( H ) $[f({W}),{W}]^{(H)}$ defined by [ f ( W ) , W ] t ( H ) : = lim ε ↓ 0 1 ν 2 ε 2 H ∫ 0 t { f ( W s + ε ) − f ( W s ) } ( W s + ε − W s ) d η s $$\bigl[f({W}),{W}\bigr]^{(H)}_{t}:=\lim_{\varepsilon\downarrow 0} \frac{1}{\nu^{2}\varepsilon^{2H}} \int_{0}^{t} \bigl\{ f({W}_{ s+\varepsilon})-f({W}_{s}) \bigr\} ({W}_{s+\varepsilon}-{W}_{s}) \,d\eta_{s} $$ in probability, where f is a Borel function and η s = λ 2 s + ν 2 s 2 H $\eta_{s}=\lambda^{2}s+\nu^{2}s^{2H}$ . For some suitable function f we show that the quadratic covariation exists in L 2 ( Ω ) $L^{2}(\Omega)$ and the Itô formula F ( W t ) = F ( 0 ) + ∫ 0 t f ( W s ) d W s + 1 2 [ f ( W ) , W ] t ( H ) $$F({W}_{t})=F(0)+ \int_{0}^{t}f({W}_{s})\,dW_{s}+ \frac{1}{2}\bigl[f({W}),{W}\bigr]^{(H)}_{t} $$ holds for all absolutely continuous function F with F ′ = f $F'=f$ , where the integral is the Skorohod integral with respect to W. |
| first_indexed | 2024-12-23T06:02:25Z |
| format | Article |
| id | doaj.art-19ff1faaed1d4b2596b6850737c51baa |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-23T06:02:25Z |
| publishDate | 2016-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-19ff1faaed1d4b2596b6850737c51baa2022-12-21T17:57:38ZengSpringerOpenJournal of Inequalities and Applications1029-242X2016-11-012016112010.1186/s13660-016-1254-2The quadratic variation for mixed-fractional Brownian motionHan Gao0Kun He1Litan Yan2Glorious Sun School of Business and Management, Donghua UniversityDepartment of Mathematics, Donghua UniversityDepartment of Mathematics, Donghua UniversityAbstract Let W = λ B + ν B H ${W}=\lambda B+\nu B^{H}$ be a mixed-fractional Brownian motion with Hurst index 0 < H < 1 2 $0< H<\frac{1}{2}$ and λ , ν ≠ 0 $\lambda,\nu\neq0$ . In this paper we study the quadratic covariation [ f ( W ) , W ] ( H ) $[f({W}),{W}]^{(H)}$ defined by [ f ( W ) , W ] t ( H ) : = lim ε ↓ 0 1 ν 2 ε 2 H ∫ 0 t { f ( W s + ε ) − f ( W s ) } ( W s + ε − W s ) d η s $$\bigl[f({W}),{W}\bigr]^{(H)}_{t}:=\lim_{\varepsilon\downarrow 0} \frac{1}{\nu^{2}\varepsilon^{2H}} \int_{0}^{t} \bigl\{ f({W}_{ s+\varepsilon})-f({W}_{s}) \bigr\} ({W}_{s+\varepsilon}-{W}_{s}) \,d\eta_{s} $$ in probability, where f is a Borel function and η s = λ 2 s + ν 2 s 2 H $\eta_{s}=\lambda^{2}s+\nu^{2}s^{2H}$ . For some suitable function f we show that the quadratic covariation exists in L 2 ( Ω ) $L^{2}(\Omega)$ and the Itô formula F ( W t ) = F ( 0 ) + ∫ 0 t f ( W s ) d W s + 1 2 [ f ( W ) , W ] t ( H ) $$F({W}_{t})=F(0)+ \int_{0}^{t}f({W}_{s})\,dW_{s}+ \frac{1}{2}\bigl[f({W}),{W}\bigr]^{(H)}_{t} $$ holds for all absolutely continuous function F with F ′ = f $F'=f$ , where the integral is the Skorohod integral with respect to W.http://link.springer.com/article/10.1186/s13660-016-1254-2mixed fractional Brownian motionMalliavin calculuslocal timefractional Itô formula |
| spellingShingle | Han Gao Kun He Litan Yan The quadratic variation for mixed-fractional Brownian motion Journal of Inequalities and Applications mixed fractional Brownian motion Malliavin calculus local time fractional Itô formula |
| title | The quadratic variation for mixed-fractional Brownian motion |
| title_full | The quadratic variation for mixed-fractional Brownian motion |
| title_fullStr | The quadratic variation for mixed-fractional Brownian motion |
| title_full_unstemmed | The quadratic variation for mixed-fractional Brownian motion |
| title_short | The quadratic variation for mixed-fractional Brownian motion |
| title_sort | quadratic variation for mixed fractional brownian motion |
| topic | mixed fractional Brownian motion Malliavin calculus local time fractional Itô formula |
| url | http://link.springer.com/article/10.1186/s13660-016-1254-2 |
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