arXiv research

We solve for functions from their truncated Hilbert transforms using Chebyshev series.

Improved numerical solution for BSDEs with reduced boundary errors.

LLMs struggle with arithmetic tasks unless they use high numerical precision.

The ability to decompose a signal in an orthonormal basis (a set of orthogonal components, each normalized to have unit length) using a fast numerical procedure rests at the heart of many signal processing methods and applications. The classic examples are the Fourier and wavelet transforms that enjoy numerically effic…

Analytical pricing formulas and Greeks are obtained for European and American basket put options using Mellin transforms. We assume assets are driven by geometric Brownian motion which exhibit correlation and pay a continuous dividend rate. A novel approach to numerical Mellin inversion is achieved via the fast Fourier…

In this paper we apply the innovative Laplace transformation method introduced by Sheen, Sloan, and Thomée (IMA J. Numer. Anal., 2003) to solve the Black-Scholes equation. The algorithm is of arbitrary high convergence rate and naturally parallelizable. It is shown that the method is very efficient for calculating vari…

In the framework of bilateral Gamma stock models we seek for adequate option pricing measures, which have an economic interpretation and allow numerical calculations of option prices. Our investigations encompass Esscher transforms, minimal entropy martingale measures, $p$-optimal martingale measures, bilateral Esscher…

In this paper, we propose several dictionary learning algorithms for sparse representations that also impose specific structures on the learned dictionaries such that they are numerically efficient to use: reduced number of addition/multiplications and even avoiding multiplications altogether. We base our work on facto…

The study analyzes numerical stability in large language models using mixed-precision arithmetic.

We present a numerical implementation of the geodesic ray transform and its inversion over functions and solenoidal vector fields on two-dimensional Riemannian manifolds. For each problem, inversion formulas previously derived in \cite{Pestov2004,Krishnan2010} are implemented in the case of simple and some non-simple m…

Random forest models systematically bias predictions; a numerical transform corrects this.

Paper presents a fast algorithm for pricing Bermudan swaptions under the two-factor Hull-White model.

A novel graph spectral method for mixed categorical and numerical data.

Twelve numerical methods for Poisson geometry concepts.

Conventional mutual information (MI) based feature selection (FS) methods are unable to handle heterogeneous feature subset selection properly because of data format differences or estimation methods of MI between feature subset and class label. A way to solve this problem is feature transformation (FT). In this study,…

Efficiently calculates Brazilian stock options with discrete dividends.

Signal models based on sparsity, low-rank and other properties have been exploited for image reconstruction from limited and corrupted data in medical imaging and other computational imaging applications. In particular, sparsifying transform models have shown promise in various applications, and offer numerous advantag…

Bayesian quadrature improves integration efficiency with invariant priors.

Invariants for surfaces up to rigid transformations, with a comeagre subset retrieval algorithm.

The paper derives formulas for option pricing and random walk expectations.

We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast F…

Transformer improves parameter estimation without needing closed-form solutions.

Efficiently approximates eigenspaces for symmetric and general matrices.

Paper derives a simplified formula for Expected Improvement using log-transformed data.

A new transform links rotating calorons to solutions of a differential equation.

A new algorithm computes Fourier coefficients for a specified range efficiently.

Computation of moments of transformed random variables is a problem appearing in many engineering applications. The current methods for moment transformation are mostly based on the classical quadrature rules which cannot account for the approximation errors. Our aim is to design a method for moment transformation for …

New efficient method for inverse Z-transform reduces complexity significantly.

Develops a new option pricing model under G-expectation framework.

Learned data models based on sparsity are widely used in signal processing and imaging applications. A variety of methods for learning synthesis dictionaries, sparsifying transforms, etc., have been proposed in recent years, often imposing useful structures or properties on the models. In this work, we focus on sparsif…

The aim of this article is to design a moment transformation for Student- t distributed random variables, which is able to account for the error in the numerically computed mean. We employ Student-t process quadrature, an instance of Bayesian quadrature, which allows us to treat the integral itself as a random variable…

We consider a semilinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity which is related to a stochastic control problem with fuel constraint. The fuel constraint translates into a singular initial condition for the HJB equation. We first propose a transformation based on a change of vari…

New method transforms complex stochastic equations into simpler ones for efficient simulation.

We derive reconstruction formulas for a family of geodesic ray transforms with connection, defined on simple Riemannian surfaces. Such formulas provide injectivity of such all transforms in a neighbourhood of constant curvature metrics and non-unitary connections with curvature close to zero. If certain Fredholm equati…

The paper studies Fourier-Laplace transforms in polynomial OU volatility models for option pricing.

The square root of Fredholm determinants causes numerical instabilities in option pricing models.

Data is said to follow the transform (or analysis) sparsity model if it becomes sparse when acted on by a linear operator called a sparsifying transform. Several algorithms have been designed to learn such a transform directly from data, and data-adaptive sparsifying transforms have demonstrated excellent performance i…

Study shows neural network parameters converge to ridgelet spectrum.

Paper presents a Transformer model for automatic domain adaptation.

We establish several closed pricing formula for various path-independent payoffs, under an exponential Lévy model driven by the Variance Gamma process. These formulas take the form of quickly convergent series and are obtained via tools from Mellin transform theory as well as from multidimensional complex analysis. Par…

The paper studies scaling laws for associative memory mechanisms.

We study the geodesic X-ray transform $X$ on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of $X$ and relate it to the con…

Improves data normality with robust transformations.

We study the integral transform over a general family of broken rays in $\mathbb{R}^2$. It is natural for broken rays to have conjugate points, for example, when they are reflected from a curved boundary. If there are conjugate points, we show that the singularities cannot be recovered from local data and therefore art…

New methods for $Z$-transform inversion and Wiener-Hopf factorization.

Develops a new method to compute risk-sharing allocations using Laplace transforms.

We illustrate how to compute local risk minimization (LRM) of call options for exponential Lévy models. We have previously obtained a representation of LRM for call options; here we transform it into a form that allows use of the fast Fourier transform method suggested by Carr & Madan. In particular, we consider Merton…

Transforms curves and surfaces for efficient geometric analysis.