2026/06/05 by Arthur Bourdon, Thibault Jeannin · 1 citation
Mathematics · #math.PR
We study two probabilistic questions for stochastic Volterra equations arising in rough volatility. These equations underlie some of the most popular non-Markovian stochastic volatility models in mathematical finance. First, we establish subcritical positive moment bounds for stochastic exponentials driven by Gaussian Volterra processes. In the Gaussian Volterra-Bergomi setting, we prove that if ρ∈[-1,0), then 𝔼[STp]<∞ for every 0<p<pρ, where p-1=∞ and pρ=(1-ρ2)-1 for -1<ρ<0. For the fractional rough Bergomi kernel, we additionally prove explosion at the critical exponent p=pρ. Combined with the known explosion above the threshold, this yields the exact criterion 𝔼[STp]<∞ if and only if 0<p<pρ in the fractional rough Bergomi model. Second, for the fractional Volterra square-root process, equivalently the rough Heston variance process, we prove that its law has a positive atom at zero at every positive time. In particular, no Feller-type condition can make the zero boundary inaccessible in the fractional rough Heston regime.