“…where K is a scale factor; -the multifractional Brownian motion (mBm) with functional parameter H (t), for which α(t, ω) = H (t) a.s.; -the Multifractional Processes with Random Exponents (MPRE) of parameter H (t, ω), for which-under some technical conditions (see Theorem 3.1 in Ayache and Taqqu ( 2005))-it is again α(t, ω) = H (t, ω) a.s.; -the symmetric α-stable Lévy motion (0 < α ≤ 2), for which H = 1/α a.s.; -the fractional Lévy motion, of parameter H − 1 2 + 1 α a.s. All these fractal processes have been considered to some extent as potential models of the financial dynamics Tapiero et al (2016); in particular, the mBm and the MPRE seem to account for many stylized facts (Bouchaud 2005), primarily the log-return heteroskedasticity, which constitutes one of the main challenges for VaR assessment. For this reason, following Costa and Vasconcelos (2003), Frezza (2012), Bianchi and Pianese (2014), Corlay et al (2014), Garcin (2017), Bertrand et al (2018)), we will assume the log-price process to be modeled as an MPRE with random parameter H (t) := H (t, ω).…”