arXiv 1609.073522016-09math.PR

关于$H>\frac{1}{2}$分数布朗运动的特征标和体积的一些结果

Some results on the Signature and Cubature of the Fractional Brownian motion for $H>\frac{1}{2}$

Riccardo Passeggeri

备注：: New results in particular on the cubature method have been added

摘要：

本文给出了分数布朗运动（fBm）的特征和体积的不同结果。第一个结果是关于fBM的线性分段近似的预期签名的收敛速度到其确切值，赫斯特参数$H\in(\frac{1}{2},1)$的值。我们证明了收敛速度由$2H$给出。我们相信这个速度是快的，因为它与Ni和Xu的结果一致，他们证明了布朗运动（即fBm与$H=\frac{1}{2}$）的快收敛速度由$1$给出。第二个结果是关于在第一个结果中得到的收敛速度的系数的界。我们得到了$\frac{\tilde{A}k(2k-1)}{(k-1)!2^{k}}$签名的第$2k$项的系数的统一界，其中$\tilde{A}$是与$k$无关的有限常数。第三个结果涉及fBm的预期特征的急剧衰减率。我们得到了$\frac{1}{k!2^{k}}$的期望签名的$2k$ -th项的一个精确界。最后的结果涉及体积法的fBm为$H>\frac{1}{2}$。特别是，我们开发的体积法的框架fBm，提供了一个界的近似误差在一般情况下，并获得体积公式的fBm在一个特定的设置。这些结果推广了Lyons和Victoir的工作，他们专注于布朗运动的情况。

英文摘要：

In this work we present different results concerning the signature and the cubature of fractional Brownian motion (fBm). The first result regards the rate of convergence of the expected signature of the linear piecewise approximation of the fBm to its exact value, for a value of the Hurst parameter $H\in(\frac{1}{2},1)$. We show that the rate of convergence is given by $2H$. We believe that this rate is sharp as it is consistent with the result of Ni and Xu, who showed that the sharp rate of convergence for the Brownian motion (i.e. fBm with $H=\frac{1}{2}$) is given by $1$. The second result regards the bound of the coefficient of the rate of convergence obtained in the first result. We obtain an uniform bound for the coefficient for the $2k$-th term of the signature of $\frac{\tilde{A}k(2k-1)}{(k-1)!2^{k}}$, where $\tilde{A}$ is a finite constant independent of $k$. The third result regards the sharp decay rate of the expected signature of the fBm. We obtain a sharp bound for the $2k$-th term of the expected signature of $\frac{1}{k!2^{k}}$. The last results concern the cubature method for the fBm for $H>\frac{1}{2}$. In particular, we develop the framework of the cubature method for fBm, provide a bound for the approximation error in the general case, and obtain the cubature formula for the fBm in a particular setting. These results extend the work of Lyons and Victoir, who focused on the Brownian motion case.

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