Weak Convergence to the Two-Parameter Volterra Multifractional Process in Besov Spaces

Abstract:

      In this paper, we prove that two-parameter Volterra multifractional process can be approximated in law in the topology of the anisotropic Besov spaces by the family of processes $\{B_{n}(s,t)\}_{n\in \mathbb{N}}$ defined by $$B_{n}(s,t)=\int_{0}^{s}\int_{0}^{t}K_{\alpha(s)}(s,u)K_{\beta(t)}(t,v)\theta_{n}(u,v)\d u\d v,$$ where $\{\theta_{n}(u,v)\}_{n\in \mathbb{N}}$ is a family of processes, converging in law to a Brownian sheet as $n\rightarrow\infty$, based on the well known Donsker's theorem.