An Optimal Approximation of Rosenblatt Sheet by Multiple Wiener Integrals

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• 作者：Guangjun Shen ; Qian Yu
• 关键词：Rosenblatt sheet ; approximation ; Wiener integrals
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：14
• 期：2
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1660-5454
• 卷排序：14

文摘

Let \(Z^{\alpha ,\beta }\) be the Rosenblatt sheet with the representation $$\begin{aligned} Z^{\alpha ,\beta }(t,s)=\int ^t_0\int ^s_0\int ^t_0\int ^s_0Q^\alpha (t,y_1,y_2)Q^\beta (s,u_1,u_2)B(\mathrm{d}y_1,\mathrm{d}u_1)B(\mathrm{d}y_2,\mathrm{d}u_2), \end{aligned}$$where B is a standard Brownian sheet, \(\frac{1}{2}<\alpha ,\beta <1\), \(Q^\alpha \) and \(Q^\beta \) are the given kernel. In this paper, we obtain an optimal approximation of Rosenblatt sheet \(Z^{\alpha ,\beta }\) based on the multiple Wiener integrals of form: $$\begin{aligned}&\int ^t_0\int ^s_0\int ^t_0\int ^s_0[k_1(y_1y_2)^{-\frac{\alpha }{2}}(u_1u_2)^{-\frac{\beta }{2}}+k_2(y_1\vee y_2)^{\frac{\alpha }{2}}(y_1\wedge y_2)^{-\frac{\alpha }{2}}|y_1-y_2|^{\alpha -1} \\&\cdot (u_1\vee u_2)^{\frac{\beta }{2}}(u_1\wedge u_2)^{-\frac{\beta }{2}}|u_1-u_2|^{\beta -1}]B(\mathrm{d}y_1,\mathrm{d}u_1)B(\mathrm{d}y_2,\mathrm{d}u_2),~~k_1,k_2\ge 0. \end{aligned}$$KeywordsRosenblatt sheetapproximationWiener integralsGuangjun Shen was supported by the National Natural Science Foundation of China (11271020), the Distinguished Young Scholars Foundation of Anhui Province (1608085J06). Qian Yu was supported by the National Natural Science Foundation of China (11501009).