Filippo Beretta | Searcharxiv

arXiv2026

A Complete Spectral Analysis of the CEV Operator with Applications to Arbitrage

We provide a complete Sturm--Liouville spectral analysis of the Constant Elasticity of Variance (CEV) operator. By transforming the corresponding Fokker--Planck operator into a generalized Laguerre operator, we explicitly characterize its self-adjoint extensions, boundary conditions, spectra, and eigenfunctions across all elasticity regimes. We then relate these spectral features to arbitrage phenomena in the CEV model, showing how boundary behavior and positive harmonic functions encode the distinction between attainable-boundary arbitrage mechanisms and strict-local-martingale bubble regimes. The result is an explicit operator-theoretic perspective on the link between CEV dynamics, no-arbitrage, and spectral theory.

math.SP↗

arXiv2023

On Benford's Law and the Coefficients of the Riemann Mapping Function for the Exterior of the Mandelbrot Set

We investigate Benford's law in relation to fractal geometry. Basic fractals, such as the Cantor set and Sierpinski triangle are obtained as the limit of iterative sets, and the unique measures of their components follow a geometric distribution, which is Benford in most bases. Building on this intuition, we aim to study this distribution in more complicated fractals. We examine the Laurent coefficients of a Riemann mapping and the Taylor coefficients of its reciprocal function from the exterior of the Mandelbrot set to the complement of the unit disk. These coefficients are 2-adic rational numbers, and through statistical testing, we demonstrate that the numerators and denominators are a good fit for Benford's law. We offer additional conjectures and observations about these coefficients. In particular, we highlight certain arithmetic subsequences related to the coefficients' denominators, provide an estimate for their slope, and describe efficient methods to compute them.

math.CV↗