From Rough to Multifractal volatility: the log S-fBM model

We introduce a family of random measures $M_{H,T} (d t)$, namely log S-fBM, such that, for $H>0$, $M_{H,T}(d t) = e^{\omega_{H,T}(t)} d t$ where $\omega_{H,T}(t)$ is a Gaussian process that can be considered as a stationary version of an $H$-fractional Brownian motion. Moreover, when $H \to 0$, one has $M_{H,T}(d t) \rightarrow {\widetilde M}_{T}(d t)$ (in the weak sense) where ${\widetilde M}_{T}(d t)$ is the celebrated log-normal multifractal random measure (MRM). Thus, this model allows us to consider, within the same framework, the two popular classes of multifractal ($H = 0$) and rough volatility ($0