An optimal approximation of Rosenblatt sheet by multiple Wiener integrals

TL;DR

This paper develops an optimal approximation method for the Rosenblatt sheet using multiple Wiener integrals, providing a new approach to approximate complex stochastic processes with explicit integral representations.

Contribution

It introduces a novel construction of multiple Wiener integrals to optimally approximate the Rosenblatt sheet, advancing the understanding of its structure and simulation.

Findings

Constructed specific multiple Wiener integrals for approximation

Derived an optimal approximation scheme for the Rosenblatt sheet

Enhanced methods for simulating complex stochastic processes

Abstract

Let $Z^{\alpha,\beta}$ be the Rosenblatt sheet with the representation $$ Z^{\alpha,\beta}(t,s)=\int^t_0\int^s_0\int^t_0\int^s_0Q^\alpha(t,y_1,y_2)Q^\beta(s,u_1,u_2)B(dy_1,du_1)B(dy_2,du_2) $$ where $B$ is a Brownian sheet, $\frac12<\alpha,\beta<1$, $Q^\alpha$ and $Q^\beta$ are the given kernel. In this paper, we contruct multiple Wiener integrals of the form \begin{align*} \int^t_0\int^s_0\int^t_0\int^s_0&[k_1(y_1,y_2)^{-\frac12\alpha}(u_1,u_2)^{-\frac12\beta}+k_2(y_1\vee y_2)^{\frac12\alpha}(y_1\wedge y_2)^{-\frac12\alpha}|y_1-y_2|^{\alpha-1}\\ &\cdot(u_1\vee u_2)^{\frac12\beta}(u_1\wedge u_2)^{-\frac12\beta}|u_1-u_2|^{\beta-1}]B(dy_1,du_1)B(dy_2,du_2),~~k_1,k_2\geq0, \end{align*} and obtain an optimal approximation of $Z^{\alpha,\beta}(t,s)$.