arXiv subjects

Ramiro Fontes

Publications and source records attributed to Ramiro Fontes.

5 recordsLinked to original sources

arXiv2026

Bakry-Emery Curvature of the Fractional Laplacian via Fractional Brownian Covariance

We study Bakry-Emery curvature for fractional Laplacian generators using a Fourier representation of the carr\'e du champ operator. For the stable generator of order gamma, the associated kernel on same-sign frequencies coincides with the covariance kernel of fractional Brownian motion with Hurst parameter equal to gamma divided by two. This observation allows the curvature inequality to be reformulated as a generalized eigenvalue problem for covariance matrices. On the one dimensional torus we analyze this matrix formulation for trigonometric polynomials. In the Cauchy case (gamma equal to one), corresponding to Brownian covariance, the eigenstructure can be computed explicitly and yields a Bakry-Emery curvature bound on the corresponding Fourier subspaces. We also study the effect of adding a confining drift to the Cauchy generator and show that the curvature spectrum undergoes a simple scalar shift. These results provide a matrix formulation of Bakry-Emery curvature for certain nonlocal operators and highlight a structural connection between fractional Laplacians and fractional Brownian covariance kernels.

math.PR↗

arXiv2026

Stochastic Calculus for Rough Fractional Brownian Motion via Operator Factorization

We develop an operator-theoretic formulation of stochastic calculus for fractional Brownian motion with Hurst parameter H in (0, 1/2). The approach is based on adjointness between stochastic integration and differentiation in the Cameron-Martin space of the driving process. For Gaussian Volterra processes, we establish a canonical factorization of fluctuations (Id - E) = delta_X Pi_X D_X, where D_X := delta_X^* is the operator-covariant derivative (adjoint of the stochastic integral), delta_X the divergence, and Pi_X the predictable projection. In the rough fractional regime, the factorization yields explicit derivative formulas for cylindrical functionals, controlled expansions of conditional expectations with O(|t-s|^{2H}) remainders, and an intrinsic identification of the Gubinelli derivative as the predictable component Pi_X D_X F. The framework extends to mixed semimartingale-rough processes, providing a unified calculus without requiring iterated integrals or signature constructions.

math.PR↗

arXiv2026

Stochastic Calculus as Operator Factorization An Operator-Covariant Derivative and Unified Representation

We present a unified operator-theoretic framework for stochastic calculus based on the factorization (Id - E)F = {\delta}_X {\Pi}_X D_X F, valid for F_T^X-measurable F in L^2({\Omega}) when the driving process X has the representation property. For a square-integrable process X with stochastic integral {\delta}_X, we define the operator-covariant derivative D_X := {\delta}_X* as the Hilbert space adjoint of {\delta}_X. Combined with predictable projection {\Pi}_X, this yields a unified Clark-Ocone representation. The operator D_X F is defined as an adjoint for all F in L^2({\Omega}), without differentiability assumptions; the representation holds when X has the predictable representation property, and reduces to the Galtchouk-Kunita-Watanabe projection when it does not. The framework requires no reproducing kernel Hilbert space or Cameron-Martin structure, and applies to non-Gaussian processes. We work out concrete examples including Brownian motion, general continuous martingales, and compensated Poisson processes.

math.PR↗

arXiv2026

An Operator Ito Formula for Volterra Gaussian Processes: The Intrinsic Bracket via Causal Derivation-Divergence Factorization

We derive an Ito-type change-of-variables formula for Volterra Gaussian processes (including fractional Brownian motion with any Hurst parameter), based on the operator factorization framework. The Ito correction is expressed as a Stieltjes integral against the energy function Gamma^X(t) := ||Pi DX_t||_H^2, which equals E[X_t^2] for centered Gaussian processes. The correction emerges from the non-commutativity of the predictable projection Pi with nonlinear functions and is computed via the Gaussian conditional expectation structure following Decreusefond-Ustunel. We prove three results beyond the formula itself: (1) the energy measure d Gamma^X is the unique second-order correction compatible with the operator factorization; (2) under a fixed driving martingale, the intrinsic bracket is invariant under changes of Volterra kernel representation; (3) the bracket is stable under L^2 kernel approximation. The proof uses a marginal density argument via the Gaussian heat equation, bypassing pathwise increments entirely. We restrict to first-order formulas; higher-order rough dynamics are not addressed. The proof relies on Gaussianity; for non-Gaussian processes the formula extends as a conjecture.

math.PR↗

arXiv2011

Applications of the quadratic covariation differentiation theory: variants of the Clark-Ocone and Stroock's formulas

In a 2006 article (\cite{A1}), Allouba gave his quadratic covariation differentiation theory for Itô's integral calculus. He defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic differentiation theory containing a fundamental theorem of stochastic calculus relating this derivative to Itô's integral, a differential stochastic chain rule, a differential stochastic mean value theorem, and other differentiation rules. Here, we use this differentiation theory to obtain variants of the Clark-Ocone and Stroock formulas, with and without change of measure. We prove our variants of the Clark-Ocone formula under $L^{2}$-type conditions; with no Malliavin calculus, without the use of weak distributional or Radon-Nikodym type derivatives, and without the significant machinery of the Hida-Malliavin calculus. Unlike Malliavin or Hida-Malliavin calculi, the form of our variant of the Clark-Ocone formula under change of measure is as simple as it is under no change of measure, and without requiring any further differentiability conditions on the Girsanov transform integrand beyond Novikov's condition. This is due to the invariance under change of measure of the first author's derivative in \cite{A1}. The formulations and proofs are natural applications of the differentiation theory in \cite{A1} and standard Itô integral calculus. Iterating our Clark-Ocone formula, we obtain variants of Stroock's formula. We illustrate the applicability of these formulas by easily, and without Hida-Malliavin methods, obtaining the representation of the Brownian indicator $F=\mathbb{I}_{[K,\infty)}(W_{T})$, which is not standard Malliavin differentiable, and by applying them to digital options in finance. We then identify the chaos expansion of the Brownian indicator.

math.PR↗