Limit theorem for a differential equation with a long-range random coefficient
The author considers a differential equation with a small parameter \(\varepsilon\) and a centered stationary long-range Gaussian process \(m\). For instance \(m\) may be a fractional white noise with Hurst index \(H\) (\(1/2<H<1\)). It is proved that the solution of this equation converges in distribution as \(\varepsilon\) goes to zero to the solution of a stochastic differential equation driven by a fractional Brownian motion: when \(m\) is an \(H\)-fractional white noise, this fractional Brownian motion has the same Hurst index. The proof is based on the theory of rough paths that has recently been put forward by T. Lyons.
scientific article; zbMATH DE number 2057080