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Lévy processes as weak limits of rough Heston models

Alessandro Bondi, Martin Forde

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 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 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 Lévy processes. This convergence result is then strengthened to a functional setting, namely on the space of càdlàg functions on the non-negative half-line endowed with the $M_1$ topology.

Lévy processes as weak limits of rough Heston models

Abstract

We show weak convergence of the time- marginals for the integrated variance in a re-scaled rough Heston model to an Inverse Gaussian Lévy process. This shows we can obtain such a limit without having to impose that the true Hurst exponent for the model is as in [Abi Jaber, & De Carvalho, 2024], or that as in [Abi Jaber, Attal, & Rosenbaum, 2025], so the result potentially has increased financial relevance. We later extend the analysis to the case where 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 Lévy processes. This convergence result is then strengthened to a functional setting, namely on the space of càdlàg functions on the non-negative half-line endowed with the topology.

Paper Structure

This paper contains 11 sections, 14 theorems, 120 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Marginal Integrated Variance Asymptotics in a Re-Scaled Rough Heston Model
  3. Adding jumps into $V^{\varepsilon}$
  4. Weak functional convergence in the $M_1$ topology
  5. Proof of Lemma \ref{['lem:E+U']}
  6. Proof of Lemma \ref{['lem:Ale']}
  7. Relative compactness in $L^1$
  8. Characterization of the limit points of $\psi_{\varepsilon}$
  9. Conclusion
  10. Laplace transform of hitting time to an upper barrier for a spectrally negative Lévy process
  11. Brief formal derivation of the main idea in AAR25

Key Result

Theorem 2.1

Consider a re-scaled rough Heston model for the variance process $V^\varepsilon=(V^\varepsilon_t)_{t\ge0}$: where $V_0\ge0$, $W$ is a standard one-dimensional Brownian motion, $\alpha\in (\frac{1}{2},1)$ and $\lambda,\sigma>0$. Then, for every $t>0$ fixed, $A^\varepsilon_t=\int_0^tV^{\varepsilon}_s ds$ tends weakly, as $\varepsilon\to 0$, to the time-$t$ marginal of an Inverse Gaussian Lévy proce

Figures (1)

  • Figure 1: Here we have plotted $\psi_{\varepsilon}$ in Eq. \ref{['eq:alt']} (in blue) and $\psi_0$ in Eq. \ref{['lim_eq']} (grey dashed), using an Adams scheme with 2000 time steps with $\varepsilon= .01$, $H =0.2$, $\nu= .4$, $\lambda = 1$, $T = 1$, $f(s) =-\frac{1}{2} (1_{\{s<\frac{1}{2}\}} +1_{\{s\le 1\}})$ and $\nu(x)=\frac{C e^{-M x}}{x^{1+Y}}1_{\{x>0\}}$ for $C=1$, $M=3$ and $Y=1.5$. Notice that such a $\nu$ is the Lévy measure of a one-sided tempered stable (CGMY) process. We see convergence to $\psi_0$ (see e.g. BL24 for details on refinements to Adams schemes). Numerically solving the VIE in Eq. \ref{['eq:Vint']} for $\varepsilon\ll 1$ appears to be much harder due to the Dirac nature of the kernel.

Theorems & Definitions (36)

  • Definition 2.1
  • Theorem 2.1
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 26 more