THE MULTIPLICATIVE CHAOS OF H=0 FRACTIONAL BROWNIAN FIELDS

• 成果类型: Article
• 署名作者: Hager, Paul; Neuman, Eyal
• 署名单位: Humboldt University of Berlin; Imperial College London
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1730
• 发表日期: 2022
• 页码: 2139-2179
• 关键词: motion; volatility

摘要

We consider a family of fractional Brownian fields {B-H} H is an element of(0,1) on R-d, where H denotes their Hurst parameter. We first define a rich class of normalizing kernels psi and we rescale the normalised field by the square-root of the gamma function Gamma(H), such that the covariance of X-H (x) = Gamma(H)(1/2) (B-H(x)- integral(Rd) B-H (u)Psi(u, x) du), converges to the covariance of a log-correlated Gaussian field when H down arrow 0. 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 M-gamma(H)(dx) = e(gamma XH) (x)-gamma(2)/2 E[X-H (x)(2)] dx, towards a Gaussian multiplicative chaos, as H down arrow 0 for all gamma is an element of(0, gamma* (d)), where gamma* (d) > root 7/4 d. As a corollary we establish the L-2 convergence of M-gamma(H) over the sets of good points, where the field X-H 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 L-2 to the volatility of multifractal random walks.