Renormalization Group Method for a Stochastic Differential Equation with Multiplicative Fractional White Noise

Abstract

In this paper, we present an application of the renormalization group method developed by Chen, Goldenfeld and Oono for a stochastic differential equation in a space of Hilbert space-valued generalized random variables with multiplicative noise. The driving process is a real-valued fractional white noise with a Hurst parameter greater than 1/2. The stochastic integration is understood in the Wick–Itô–Skorohod sense. This article is a generalization of results of Glatt-Holtz and Ziane, which were for the systems with white noise. We firstly demonstrate the process of formulating the renormalization group equation and the asymptotic solution. Then, we give rigorous proof of the consistency of the approximate solution. In addition, some numerical comparisons are given to illustrate the validity of our results.