Moments in Rough Bergomi and Boundary Attainment in Rough Heston

Abstract

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 $\rho \in [-1,0)$, then $\mathbb{E}[S_T^p]<\infty$ for every $0<p<p_{\rho }$, where $p_{-1}=\infty$ and $p_{\rho }=(1-{\rho }^2{)}^{-1}$ for $-1<\rho <0$. For the fractional rough Bergomi kernel, we additionally prove explosion at the critical exponent $p=p_{\rho }$. Combined with the known explosion above the threshold, this yields the exact criterion $\mathbb{E}[S_T^p]<\infty$ if and only if $0<p<p_{\rho }$ 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.