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arXiv 1603.01720math.PR
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加权分式布朗运动的二次协变差

The quadratic covariation for a weighted fractional Brownian motion

XIchao Sun, Litan Yan, Qinghua Zhang

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摘要：

设$B^{a,b}$是加权分数布朗运动，其指标$a,b$满足$a>-1,-1<b<0,|b|<1+a$ .本文利用$s>0$的渐近性质$$ E[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2] =O(\varepsilon^{1+b})\not\sim \varepsilon^{1+a+b}=E[(B^{a,b}_{\varepsilon})^2]\qquad (\varepsilon\to 0) $$，考虑了由$$ \bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t=\lim_{\varepsilon\downarrow 0}\frac{1+a+b}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} \left\{f(B^{a,b}_{s+\varepsilon}) -f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds, $$定义的广义二次协变$\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}$，只要极限按概率一致存在.我们构造了一个可测函数的Banach空间${\mathscr H}$，使得广义二次协变存在于$L^2(Ω)$中，广义Bouleau-Yor恒等式$$ [f(B^{a,b}),B^{a,b}]^{(a,b)}_t=-\frac1{(1+b){\mathbb B}(a+1,b+1)} \int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t) $$对所有$f\in {\mathscr H}$成立，其中${\mathscr L}^{a,b}(x,t)=\int_0^tδ(B^{a,b}_s-x)ds^{1+a+b}$是$B^{a,b}$的加权局部时，${\mathbb B}(\cdot,\cdot)$是Beta函数.

英文摘要：

Let $B^{a,b}$ be a weighted fractional Brownian motion with indices $a,b$ satisfying $a>-1,-1<b<0,|b|<1+a$. In this paper, motivated by the asymptotic property $$ E[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2] =O(\varepsilon^{1+b})\not\sim \varepsilon^{1+a+b}=E[(B^{a,b}_{\varepsilon})^2]\qquad (\varepsilon\to 0) $$ for all $s>0$, we consider the generalized quadratic covariation $\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}$ defined by $$ \bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t=\lim_{\varepsilon\downarrow 0}\frac{1+a+b}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} \left\{f(B^{a,b}_{s+\varepsilon}) -f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds, $$ provided the limit exists uniformly in probability. We construct a Banach space ${\mathscr H}$ of measurable functions such that the generalized quadratic covariation exists in $L^2(Ω)$ and the generalized Bouleau-Yor identity $$ [f(B^{a,b}),B^{a,b}]^{(a,b)}_t=-\frac1{(1+b){\mathbb B}(a+1,b+1)} \int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t) $$ holds for all $f\in {\mathscr H}$, where ${\mathscr L}^{a,b}(x,t)=\int_0^tδ(B^{a,b}_s-x)ds^{1+a+b}$ is the weighted local time of $B^{a,b}$ and ${\mathbb B}(\cdot,\cdot)$ is the Beta function.

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