L\'evy processes as weak limits of rough Heston models

TL;DR

This paper proves that the integrated variance in a scaled rough Heston model converges weakly to an Inverse Gaussian Lévy process, broadening the model's financial applicability without restrictive Hurst exponent assumptions.

Contribution

It establishes weak convergence of the integrated variance to Lévy processes in rough Heston models without specific Hurst exponent constraints, including jump processes and functional convergence.

Findings

Weak convergence to Inverse Gaussian Lévy process.

Extension to jump processes and general Lévy processes.

Functional convergence in the M1 topology.

Abstract

We show weak convergence of the time-$t$ marginals for the integrated variance in a re-scaled rough Heston model to an Inverse Gaussian L\'{e}vy 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 the analysis to the case where $V$ has jumps, showing 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 L\'evy processes. This convergence result is then strengthened to a functional setting, namely on the space of c\`adl\`ag functions on the non-negative half-line endowed with the $M_1$ topology.