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We introduce a unified stochastic framework for modeling multiscale financial volatility based on the Log Stationary Fractional Brownian Motion (Log S-fBM) model. This construction provides a continuous interpolation between multifractal volatility regimes and rough volatility dynamics, thereby capturing key empirical features observed in financial time series. We develop a statistically robust Generalized Method of Moments (GMM) estimation procedure within the small intermittency regime. Empirical findings indicate that market indices exhibit pronounced roughness, whereas individual assets display dynamics closer to the multifractal limit which is reproduced by the Nester Stationary fractional Factor model we proposed. The framework of the Log S-fBM is further extended to a multivariate setting, enabling the joint modeling of correlated assets through a multidimensional Log S-fBM structure. This extension preserves marginal properties while incorporating cross-asset dependencies, providing a coherent explanation for the observed discrepancy between index-level and single-asset volatility behavior. In addition, we propose an efficient simulation methodology for Volterra-type processes based on Random Fourier Features (RFF) approximations of the kernel with a particular focus on the S-fBM kernel. This approach yields improved numerical stability and computational efficiency, supported by theoretical error bounds and empirical validation. Overall, the proposed framework offers a consistent and tractable approach to linking rough volatility, multifractal scaling, and factor-based structures, with both theoretical and practical implications for financial modeling.

Le miroir d’un entier n dans une  base b donnée est l’entier obtenu en inversant l’ordre  des chiffres de n dans cette base.
Par exemple 31 est le miroir de 13. Dans cet exposé nous montrons qu’il existe une infinité de nombres premiers p dont le miroir dans une base donnée est un entier avec peu de facteurs premiers. Il s’agit d’un travail en commun avec Joël Rivat et Cathy Swaenepoel.