Lévy processes as weak limits of rough Heston models

We show weak convergence of the marginals for a re-scaled rough Heston model to a Normal Inverse Gaussian (NIG) Lévy process. This shows we can obtain such a limit without having to impose that the true Hurst exponent $H$ for the model is $\frac{1}{2}$ as in [Abi Jaber, & De Carvalho, 2024], or that $H\searrow -\frac{1}{2}$ as in [Abi Jaber, Attal, & Rosenbaum, 2025], so the result potentially has increased financial relevance. We later extend to the case when $V$ has jumps, where we show weak convergence of the finite-dimensional distributions of the integrated variance to a deterministic time-change of the first passage time process to lower barriers for a more general class of spectrally positive Lévy processes.