From Hyper Roughness to Jumps as $H \to -1/2$

TL;DR

The paper analyzes the weak limit of the hyper-rough square-root process as the Hurst index approaches $-\tfrac{1}{2}$, revealing a transition from continuous to jump dynamics. By working directly with the non-Markovian Volterra system, it proves joint weak convergence of $(X^n,M^n)$ to an Inverse Gaussian Lévy process $Y$ and a shifted version $(1+\lambda)Y-G_0$, in the product topology $M_1\otimes S$. The authors develop a refined topological framework to accommodate non-Skorokhod limits, leveraging Riccati-Volterra equations to characterize the limit via the convergence of characteristic functionals. The results establish a continuum between hyper-rough continuous models and jump processes and provide numerical demonstrations via the iVi scheme. This contributes a rigorous link between fractional kernel limits and pure-jump Lévy dynamics with explicit limiting parameters and representations.

Abstract

We investigate the weak limit of the hyper-rough square-root process as the Hurst index $H$ goes to $-1/2\,$. This limit corresponds to the fractional kernel $t^{H - 1 / 2}$ losing integrability. We establish the joint convergence of the couple $(X, M)\,$, where $X$ is the hyper-rough process and $M$ the associated martingale, to a fully correlated Inverse Gaussian Lévy jump process. This unveils the existence of a continuum between hyper-rough continuous models and jump processes, as a function of the Hurst index. Since we prove a convergence of continuous to discontinuous processes, the usual Skorokhod $J_1$ topology is not suitable for our problem. Instead, we obtain the weak convergence in the Skorokhod $M_1$ topology for $X$ and in the non-Skorokhod $S$ topology for $M$.

Figures (3)

• Figure 1: Convergence of $(X^n\,, M^n)_{n\geq 0}$ to a jump process as $H^n$ goes to $-1/2\,$. Top: trajectories of $\left(X^n\right)_{n \geq 0}$ (plain curves) and trajectory of the IG process $Y$ with parameters $\left(\frac{g_0}{1 + \lambda}\,, \frac{g_0^2}{\nu^2}\right)$ (black dotted curve) using the same random seed. Middle: trajectories of $\left(M^n\right)_{n \geq 0}$ (plain curves) and trajectory of the shifted IG process $(1 + \lambda)\, Y - G_0$ (black dotted curve) using the same random seed. Bottom: trajectories of $\left(G_0^n + M^n - (1 + \lambda)\,X^n\right)_{n \geq 0}\,$.
• Figure 2: Top row: Empirical density of $X^n_T$ for different values of $H^n\,$, compared with the theoretical limiting Inverse Gaussian distribution (black line). Middle row: Empirical density of $M^n_T$ for different values of $H^n\,$, compared with the theoretical limiting shifted Inverse Gaussian distribution (black line). Bottom row: Empirical joint characteristic function $\mathbb{E}\left[\exp\left(u\,X^n_T + v\,M^n_T\right)\right]$ (dotted curves), compared with the theoretical limiting Inverse Gaussian - Shifted Inverse Gaussian characteristic function (plain lines), as a function of $v\,$, for different values of $u\,$.
• Figure 3: Example of $\varepsilon$-tube (red dashed lines) for the function $\mathbbm{1}_{[1/2\,, 1]}$ (blue plain lines) for different topologies. Left: Uniform topology. Middle:$J_1$ topology. Right:$M_1$ topology.

Theorems & Definitions (35)

• Definition 2.1: Inverse Gaussian random variable
• Definition 2.2: Inverse Gaussian process
• Lemma 2.3
• Lemma 2.4
• proof
• Remark 2.5
• Remark 2.6
• Theorem 3.1
• proof
• Lemma 3.2