Approximating the Rosenblatt process by multiple Wiener integrals

Abstract

Let $Z^{H}$ be the Rosenblatt process with the representation$$Z^H_t=\int_0^t\int_0^tL^H(t,s,r)dB_{s}dB_{r},$$where $B$ is a standard Brownian motion, $\frac12<H<1$ and $L^H$ is a given kernel. By reviewing the kernel $L^H$ we construct its approximation of multiple Wiener integrals of the form $$\int_{0}^{t}\int_{0}^{t}\left\{k_{1}(sr)^{-\frac{1}{2}H} +k_{2}(s\vee r)^{\frac{1}{2}H}(s\wedge r)^{-\frac{1}{2}H}|s-r|^{H-1}\right\}dB_{s}dB_{r}, \;\;k_1,k_2\geq 0.$$We find an optimal approximation of $Z^{H}$ via calculating accurately the values of $k_1,k_2$.