arXiv research

The aim of this paper is two-fold: first, we look at the fractional Laplacian and the conformal fractional Laplacian from the general framework of representation theory on symmetric spaces and, second, we construct new boundary operators with good conformal properties that generalize the fractional Laplacian using an e…

We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.

Study connects weighted isoperimetric problems to nonlocal elliptic operator extensions.

We extend neural networks with fractional and mixed activation functions for better function approximation.

We describe a new interpretation of the fractional GJMS operators as generalized Dirichlet-to-Neumann operators associated to weighted GJMS operators on naturally associated smooth metric measure spaces. This gives a geometric interpretation of the Caffarelli--Silvestre extension for $(-Δ)^γ$ when $γ\in(0,1)$, and both…

The paper extends Merton model to price equity warrants under subdiffusive fractional Brownian motion of the short rate.

Paper introduces new fractional Dirac operator and Q-curvature.

This chapter is an attempt to present a mathematical theory of compound fractional Poisson processes. The chapter begins with the characterization of a well-known Lévy process: The compound Poisson process. The semi-Markov extension of the compound Poisson process naturally leads to the compound fractional Poisson proc…

The continuous observation of the financial markets has identified some stylized facts which challenge the conventional assumptions, promoting the born of new approaches. On the one hand, the long-range dependence has been faced replacing the traditional Gauss-Wiener process (Brownian motion), characterized by stationa…

A new distribution family extends the $\alpha$-stable distribution with a degree of freedom parameter.

Unified treatment of two extension problems using heat equation in Heisenberg group.

A new method calculates fractional moments using the moment-generating function.

The purpose of this paper is to indicate that the recently proposed Momentum fractional least mean squares (mFLMS) algorithm has some serious flaws in its design and analysis. Our apprehensions are based on the evidence we found in the derivation and analysis in the paper titled: \textquotedblleft \textit{Momentum frac…

L2O-CFGD meta-learns hyperparameters for FGD, improving performance.

The long-term dependence of Bitcoin (BTC), manifesting itself through a Hurst exponent $H>0.5$, is exploited in order to predict future BTC/USD price. A Monte Carlo simulation with $10^4$ geometric fractional Brownian motion realisations is performed as extensions of historical data. The accuracy of statistical inferen…

Fractional porous media equations yield q-Gaussian solutions for stock price returns.

Deep learning improves Hurst parameter estimation for fractional processes.

New algorithms solve word and conjugacy problems in braid group B3.

Python package for estimating Hurst exponent in fBm.

In this paper we show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram zeta function. Inspired by Minakshisundaram's ideas, we find a precise pointwise d…

Study simulates liquidity in fractional ownership markets using ABM.

The FSRM uses a multifractional process to capture price multifractality, revealing serial information for forecasting.

Herein, we propose a spatio-temporal extension of RBFNN for nonlinear system identification problem. The proposed algorithm employs the concept of time-space orthogonality and separately models the dynamics and nonlinear complexities of the system. The proposed RBF architecture is explored for the estimation of a highl…

The paper models cryptocurrency price and volatility with jumps and fractional volatility.

We give a definition of the fractional Laplacian on some noncompact manifolds, through an extension problem introduced by Caffarelli-Silvestre. While this definition in the compact case is straightforward, in the noncompact setting one needs to have a precise control of the behavior of the metric at infinity and geomet…

Composite likelihood inference of fractional Gaussian processes with sequentially optimal subset selection

New conjectures link SU(r) Vafa-Witten invariants to Ramanujan's continued fractions.

We address the problem of computing approximate marginals in Gaussian probabilistic models by using mean field and fractional Bethe approximations. As an extension of Welling and Teh (2001), we define the Gaussian fractional Bethe free energy in terms of the moment parameters of the approximate marginals and derive an …

The study compares differencing methods for financial data and finds fractional differencing improves model performance.

We extend the well-known result that any $f \in W^{1,n}(Ω,\mathbb{R}^n)$, $Ω\subset \mathbb{R}^n$ with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces $W^{s,\frac{n}{s}}(Ω)$ for any $s \geq \frac{n}{n+1}$, where the sign condition on the Jacobian is understood in a distr…

We study the structure of abelian extensions of the group $L_qG$ of $q$-differentiable loops (in the Sobolev sense), generalizing from the case of central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are…

Fine shape theory extends strong shape to noncompact metrizable spaces.

This study uses moving average cluster entropy to analyze financial market dynamics.

We consider the class of measurable functions defined in all of $\mathbb{R}^n$ that give rise to a nonlocal minimal graph over a ball of $\mathbb{R}^n$. We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation. This estimate, together with previously known…

Lazy, perfectly informed investors trade infrequently due to costs.

Modeling joint log-volatility dynamics with multivariate fractional Ornstein-Uhlenbeck process.

In this work we present a new approach on studying dynamical systems. Combining the two ways of expressing the uncertainty, using probabilistic theory and credibility theory, we have research the generalized fractional hybrid equations. We have introduced the concepts of generalized fractional Wiener process, generaliz…

Develops a bi-variate stochastic framework to model mortality and interest rates with long-range dependence.

Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric data for such models are encoded into (fractional) bi-Hamiltonian structures and as…

Robust PCA, the problem of PCA in the presence of outliers has been extensively investigated in the last few years. Here we focus on Robust PCA in the column sparse outlier model. The existing methods for column sparse outlier model assumes either the knowledge of the dimension of the lower dimensional subspace or the …

Proposes a new model for equity options calibration.

Introduces fractional k-dimensional measure bridging fractional length and area.

The theory of derivative of noninteger order goes back to Leibniz, Liouville and Riemann. Derivatives of fractional order have found many applications in recent studies in mechanics, physics, economics. In this paper we define the fractional tangent bundle on a manifold, using a method of Radu Miron. The fractional Lei…

Let $S_g$ be a closed orientable surface of genus $g \geq 2$ and $C$ a simple closed nonseparating curve in $F$. Let $t_C$ denote a left handed Dehn twist about $C$. A \textit{fractional power} of $t_C$ of \textit{exponent} $\fraction{\ell}{n}$ is an $h \in \Mod(S_g)$ such that $h^n = t_C^{\ell}$. Unlike a root of a $t…

We formulate the fractional Ricci flow theory for (pseudo) Riemannian geometries enabled with nonholonomic distributions defining fractional integro-differential structures, for non-integer dimensions. There are constructed fractional analogs of Perelman's functionals and derived the corresponding fractional evolution …

Using Caputo fractional derivative of order $α$ we build the fractional jet bundle of order $α$ and its main geometrical structures. Defined on that bundle, some fractional dynamical systems with applications to economics are studied.

Predict stock prices using HMMs trained on fractional price changes and intraday highs/ lows.

Volatility roughness studied using fractional noise-driven models.