The quadratic variation for mixed-fractional Brownian motion

• Main Authors: Han Gao, Kun He, Litan Yan
• Format: Article
• Language: English
• Published: SpringerOpen 2016-11-01
• Series: Journal of Inequalities and Applications
• Subjects: mixed fractional Brownian motion; Malliavin calculus; local time; fractional Itô formula
• Online Access: http://link.springer.com/article/10.1186/s13660-016-1254-2

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.