We consider a family of fractional Brownian fields on , where H denotes their Hurst parameter. We first define a rich class of normalizing kernels ψ and we rescale the normalised field by the square-root of the gamma function , such that the covariance of
converges to the covariance of a log-correlated Gaussian field when . We then use Berestycki’s “good points” approach (Electron. Commun. Probab. 22 (2017) Paper No. 27) in order to derive the convergence of the exponential measure of the fractional Brownian field
towards a Gaussian multiplicative chaos, as for all , where . As a corollary we establish the convergence of over the sets of “good points”, where the field has a typical behaviour. As a by-product of the convergence result, we prove that for log-normal rough volatility models with small Hurst parameter, the volatility process is supported on the sets of “good points” with probability close to 1. Moreover, on these sets the volatility converges in to the volatility of multifractal random walks.
We are very grateful to Nathanael Berestycki whose numerous useful comments enabled us to significantly improve this paper. We are also thankful to Masaaki Fukasawa, to the Associate Editor and to the anonymous referee for careful reading of the manuscript, and for a number of useful suggestions.
"The multiplicative chaos of fractional Brownian fields." Ann. Appl. Probab. 32 (3) 2139 - 2179, June 2022. https://doi.org/10.1214/21-AAP1730