arXiv subjects

Elie Attal

Publications and source records attributed to Elie Attal.

8 recordsLinked to original sources

arXiv2026

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

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 at zero. 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 $M$.

math.PR↗

arXiv2026

Affine Structure of the Brownian Signature

We establish an infinite-dimensional affine transform theory for the time-augmented Brownian signature. Our first main result shows that, for a suitable class of linear functions of the signature, the conditional Fourier-Laplace transform admits an entire signature expansion. We prove that the associated coefficients solve an infinite-dimensional linear differential equation on the extended tensor algebra. Our second main result shows that the logarithm admits a local signature expansion whose coefficients satisfy a Riccati equation on the extended tensor algebra, revealing a generalized affine structure of the Brownian signature in a genuinely path-dependent setting. In contrast to conventional affine processes, we show that this representation is intrinsically local: zeros of the Fourier-Laplace transform in the complex plane prevent any global expansion. To recover global representations, we introduce a new class of randomized Riccati equations with path-dependent terminal conditions through a recentering argument. Furthermore, we establish uniqueness of solutions to the linear and Riccati equations within a suitable class of solutions. Our results provide a theoretical framework for transform methods in non-Markovian settings, with applications to the computation of conditional distributions.

math.PR↗

arXiv2026

Stochastic control with signatures via Riccati equations on the tensor algebra

We solve in semi-explicit form a class of non-Markovian stochastic optimal control problems with path-dependent rewards, using path signatures. We reformulate the control problem as the computation of Laplace transforms of signature functionals thanks to the Boué-Dupuis representation. Exploiting recent signature representations of such transforms on tensor algebras, we determine the value process and the optimal control through an infinite-dimensional system of Riccati equations on the extended tensor algebra. We establish an explicit feedback representation of the optimal control and the value process as an infinite linear combination of the time-extended signature of the controlled process, with time-dependent coefficients. The expansions being intrinsically local, we propose a dynamic recentering algorithm to ensure a global representation over the entire time horizon. We illustrate the approach on genuinely path-dependent, non-linear examples that go beyond the tractable linear-quadratic setting, including the tracking of linear functionals of the signature and signature lifts of Volterra control problems.

math.OC↗

arXiv2026

Volterra clocks and their pure-jump limits: hitting times of curved boundaries

We introduce a class of continuous Volterra processes, called Volterra clocks, and study their singular limit as the memory kernel collapses to a Dirac mass at zero. The dynamics are parametrised by a function $f$ acting as a nonlinear time-change, generalising the Volterra square-root process and recovering it when $f$ is affine. In the singular limit, the continuous Volterra clock converges weakly to a pure-jump process given by first passage times of a Brownian motion to curved boundaries, including affine and square-root boundaries when $f$ is, respectively, affine or quadratic. Outside the affine setting, characteristic function methods are no longer available, and we instead identify the limit directly from the dynamics. We do this through a topological framework adapted to the time-change structure which involves Skorokhod's $M_1$ topology and a decorated notion of convergence. Our analysis unifies several regimes of interest for general Volterra clocks, including large-time asymptotics, fast mean reversion, and hyper-roughness. In particular, this subsumes and extends existing results in the affine setting.

math.PR↗

arXiv2025

Efficient Simulation of Hawkes Processes using their Affine Volterra Structure

We introduce a novel and efficient simulation scheme for Hawkes processes on a fixed time grid, leveraging their affine Volterra structure. The key idea is to first simulate the integrated intensity and the counting process using Inverse Gaussian and Poisson distributions, from which the jump times can then be easily recovered. Unlike conventional exact algorithms based on sampling jump times first, which have random computational complexity and can be prohibitive in the presence of high activity or singular kernels, our scheme has deterministic complexity which enables efficient large-scale Monte Carlo simulations and facilitates vectorization. Our method applies to any nonnegative, locally integrable kernel, including singular and non-monotone ones. By reformulating the scheme as a stochastic Volterra equation with a measure-valued kernel, we establish weak convergence to the target Hawkes process in the Skorokhod $J_1$-topology. Numerical experiments confirm substantial computational gains while preserving high accuracy across a wide range of kernels, with remarkably improved performance for a variant of our scheme based on the resolvent of the kernel.

math.PR↗

arXiv2025

Simulating integrated Volterra square-root processes and Volterra Heston models via Inverse Gaussian

We introduce a novel simulation scheme, iVi (integrated Volterra implicit), for integrated Volterra square-root processes and Volterra Heston models based on the Inverse Gaussian distribution. The scheme is designed to handle $L^1$ kernels with singularities by relying solely on integrated kernel quantities, and it preserves the non-decreasing property of the integrated process. We establish weak convergence of the iVi scheme by reformulating it as a stochastic Volterra equation with a measure kernel and proving a stability result for this class of equations. Numerical results demonstrate that convergence is achieved with very few time steps. Remarkably, for the rough fractional kernel, unlike existing schemes, convergence seems to improve as the Hurst index $H$ decreases and approaches $-1/2$.

q-fin.MF↗

arXiv2025

Eigenvector Overlaps of Random Covariance Matrices and their Submatrices

We consider the singular vectors of any $m \times n$ submatrix of a rectangular $M \times N$ Gaussian matrix and study their asymptotic overlaps with those of the full matrix, in the macroscopic regime where $N \,/\, M\,$, $m \,/\, M$ as well as $n \,/\, N$ converge to fixed ratios. Our method makes use of the dynamics of the singular vectors and of specific resolvents when the matrix coefficients follow Brownian trajectories. We obtain explicit forms for the limiting rescaled mean squared overlaps for right and left singular vectors in the bulk of both spectra, for any initial matrix $A\,$. When it is null, this corresponds to the Marchenko-Pastur setup for covariance matrices, and our formulas simplify into Cauchy-like functions.

math.PR↗

arXiv2024

Interlacing Eigenvectors of Large Gaussian Matrices

We consider the eigenvectors of the principal minor of dimension $n< N$ of the Dyson Brownian motion in $\mathbb{R}^{N}$ and investigate their asymptotic overlaps with the eigenvectors of the full matrix in the limit of large dimension. We explicitly compute the limiting rescaled mean squared overlaps in the large $n\,, N$ limit with $n\,/\,N$ tending to a fixed ratio $q\,$, for any initial symmetric matrix $A\,$. This is accomplished using a Burgers-type evolution equation for a specific resolvent. In the GOE case, our formula simplifies, and we identify an eigenvector analogue of the well-known interlacing of eigenvalues. We investigate in particular the case where $A$ has isolated eigenvalues. Our method is based on analysing the eigenvector flow under the Dyson Brownian motion.

math.PR↗