Automated translation of mathematical formulae using recursive neural networks.
problem Performing translations between different representations of mathematical formulae.
method Recursive neural networks with multi-variate multi-valued Long Short-Term Memory cells and novel clustering and mini-batch training techniques.
result Achieved a prediction accuracy of 47.05% for predicting symbols at the correct position and 92.3% when ignoring the predicted position.
Extends Itô's formula for path-dependent functions in finance.
problem Modeling and hedging of path-dependent financial options.
method Functional extension of Itô's formula for C^{0,1}-functions of continuous weak Dirichlet processes.
result Validates the hedging or superhedging problems for path-dependent options.
Mathematical formulas for elliptic curve integrals solve anomaly equations.
problem Mathematical formulation of contact term singularities on elliptic curves.
method Residue formulas and holomorphic anomaly equations.
result Regularized integrals on elliptic curves satisfy holomorphic anomaly equations.
Graph neural networks can perform approximate reasoning in latent space for mathematical statements.
problem Can neural networks perform steps of approximate reasoning in a fixed dimensional latent space?
method Design and conduct an experiment using graph neural networks to predict rewrite-success of mathematical statements in a latent space.
result Graph neural networks can make non-trivial predictions about rewrite-success of statements in latent space.
Analyzes packing of circles in bounded and unbounded planes using mathematical formulas.
problem Finding optimal radii for packing circles in various plane regions.
method Deterministic analytic formulae and recurrence relations.
result Formulated analytic formulae for 2D circle packing on various plane shapes.
Proves an Euler-type formula for Möbius strip partitions.
problem No specific problem stated; focuses on a mathematical formula.
method Analyzes partitions of the Möbius strip.
result Proves an Euler-type formula for Möbius strip partitions.
The BBF, SABR, and rough SABR formulas provide nearly arbitrage-free implied vol approximations.
problem Arbitrage in implied volatility calculations.
method Analytical proofs for BBF, SABR, and rough SABR formulas under specific models.
result These formulas offer asymptotically arbitrage-free approximations of implied volatility.
Paper proves integration by parts formula for specific mathematical product.
problem Developing integration by parts formula for non-pluripolar product.
method Generalization of previous results for potentials with small unbounded loci.
result Integration by parts formula for non-pluripolar product on compact Kähler manifolds.
Abstract mathematical formulas for statistical structures and curvatures.
problem Developing formulas for statistical structures and curvatures.
method Proving new formulas and theorems for statistical structures and curvatures.
result Generalized formulas for statistical structures and curvatures.
Self-supervised skip-tree training improves mathematical reasoning in language models.
problem Improving logical reasoning in language models for formal mathematics.
method Self-supervised language modeling on mathematical formulas, skip-tree task.
result Models trained on skip-tree task outperform standard models in mathematical reasoning tasks.
Proves a general connected sum formula for families Seiberg-Witten invariants.
problem Limited connected sum formulae for families Seiberg-Witten theory.
method Develops a general connected sum formula incorporating previous results.
result Proves a new connected sum formula for Seiberg-Witten families.
Golden age of mathematical finance in the late 20th century.
problem Foundations of mathematical finance during the late 20th century.
method Collaboration between economists and probabilists.
result Established two fundamental theorems of arbitrage theory and close formulas for options.
The inversion formula for conservative multifractal measures was unveiled mathematically a decade ago, which is however not well tested in real complex systems. In this Letter, we propose to verify the inversion formula using high-frequency turbulent financial data. We construct conservative volatility measure based on…
We establish an explicit pricing formula for the class of Lévy-stable models with maximal negative asymmetry (Log-Lévy model with finite moments and stability parameter 1<α≤2) in the form of rapidly converging series. The series is obtained with help of Mellin transform and the residue theory in C2. T…
Geometric proof of Regge symmetry in different geometries.
problem Regge symmetry in tetrahedra edge lengths.
method Simple geometric proof in Euclidean, spherical, and hyperbolic geometries.
result Verification of Regge symmetry across different geometries.
Following a hedging based approach to model free financial mathematics, we prove that it should be possible to make an arbitrarily large profit by investing in those one-dimensional paths which do not possess local times. The local time is constructed from discrete approximations, and it is shown that it is α-Hölder …
The paper synthesizes the mathematics of modeling the future.
problem Modeling the future
method Unified mathematical synthesis
result Explicit connection of classical objects into a unified forecasting calculus
Paper derives formulas for surface variations in shell theory.
problem Deriving first variation formulas for surfaces in thin shell theory.
method Using strain-displacement relations from thin shell theory.
result Provides formulas for linear Weingarten surfaces as stationary points.
New proof and formula linking fusion trees to quantum knot invariants.
problem Quantum knot invariants encoding in non-semisimple TQC.
method Connection between fusion trees and Lawrence representations, using graphical calculus.
result Explicit encoding of quantum knot invariants via fusion trees.
We study eta-invariants on odd dimensional manifolds with boundary. The dependence on boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of the (inverse) determinant line of the boundary. We prove a gluing law and a variation formula for this invariant. This yields a new, …
Introduces new mathematical concepts and applies them to cohomology calculations.
problem Cohomology calculations on smooth and complex manifolds.
method Introduces new sheaf concepts (cdp presheaf, cds precosheaf) and applies Mayer-Vietoris systems.
result Explicit blow-up formulas for cohomologies, comparison with Rao et al. formula.
Principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in combinatorial sense). We express rational local formulas for all…
The paper defines the time function of stock prices using a mathematical model.
problem Understanding the movement and predictability of stock prices over time.
method Empirical evidence and mathematical modeling of white noise.
result Derives auto-correlation function, displacement formula, and power spectral density of stock price movement.
Study Gram determinants in knot theory, focusing on a Möbius band determinant.
problem Closed formula for the Gram determinant of type (Mb)1. method Survey of Gram determinants, focusing on a Möbius band determinant.
result Speculation on closed formula for (Mb)1 Gram determinant. The paper proves formulas for determinant determinants of Laplacians on Riemann surfaces with conical singularities.
problem Determinants of Laplacians on Riemann surfaces with conical singularities.
method Polyakov-Alvarez type comparison formulas for determinants of Friedrichs extensions of Laplacians.
result Determine how determinants depend on conical singularities and provide explicit formulas.
Based on criteria of mathematical simplicity and consistency with empirical market data, a stochastic volatility model is constructed, the volatility process being driven by fractional noise. Price return statistics and asymptotic behavior are derived from the model and compared with data. Deviations from Black-Scholes…
The main purpose of this paper is to formalize the modelling process, analysis and mathematical definition of corruption when entering into a contract between principal agent and producers. The formulation of the problem and the definition of concepts for the general case are considered. For definiteness, all calculati…
Study on deep neural networks using branching processes and Mehler's formula.
problem Understanding the mathematical role of activation functions in compositional neural networks.
method Connection between compositional kernels and branching processes via Mehler's formula; new random features algorithm.
result Explicit formulas for eigenvalues of compositional kernels quantify complexity.
The abstract reviews financial concepts using physics.
problem Financial pricing and risk management.
method Discrete time formalism, path integral, Green's function formulas.
result Formulas for pricing and risk mitigation methods.
New formula and algorithm for computing distances on complex Riemann surfaces.
problem Computing distances on higher-genus Riemann surfaces is challenging due to infinite terms in the formula.
method Derived a computable distance formula and developed an efficient algorithm.
result Reduced distance computation from an infimum to a minimum over a finite set of terms.
Extends Clark-Ocone theorem to non-Malliavin differentiable random variables using Ito's formula.
problem Extending Clark-Ocone theorem to non-Malliavin differentiable random variables.
method Uses Ito's formula instead of Malliavin calculus.
result Explicit representation of locally risk-minimizing strategies for digital options in Levy models.
The Black-Scholes model (sometimes known as the Black-Scholes-Merton model) gives a theoretical estimate for the price of European options. The price evolution under this model is described by the Black-Scholes formula, one of the most well-known formulas in mathematical finance. For their discovery, Merton and Scholes…
A new formula connects supersymmetric path integrals to Chern-Simons theory.
problem Constructing a rigorous path integral for supersymmetric theories on spin manifolds.
method Using Chen differential forms and non-commutative geometry, a Chern-Simons transgression formula is derived.
result The supersymmetric path integral induces a differential topological invariant.
Framework for verifying deep learning operators.
problem Complexity and errors in designing and implementing custom operators.
method Symbolic execution, syntax-guided synthesis, SMT-based verification.
result Effective synthesis and verification of deep learning operators.
RiskMiner discovers formulaic alphas using MCTS for better performance.
problem Mining formulaic alphas without considering structural information and alpha correlations.
method Formulates alpha mining as an MDP and solves it with a risk-seeking MCTS.
result Our method outperforms state-of-the-art benchmarks and achieves the most profitable results.
Characterization of the American put option price is still an open issue. From the beginning of the nineties there exists a non-closed formula for this price but nontrivial numerical computations are required to solve it. Strong efforts have been done to propose methods more and more computationally efficient but most …
The paper provides formulas linking knot invariants to deformation quantization.
problem Deformation quantization of the space of connections on a 2-manifold.
method Using Chern-Simons gauge theory in 3D, the paper derives explicit formulas for star products.
result Explicit formulas connect knot invariants to deformation quantization and gauge theory.
We provide a surprising new application of classical approximation theory to a fundamental asset-pricing model of mathematical finance. Specifically, we calculate an analytic value for the correlation coefficient between exponential Brownian motion and its time average, and we find the use of divided differences greatl…
The article prices exchange options using variance gamma-like models.
problem Pricing exchange options under specific stochastic processes.
method Derives formulas for variance gamma and variance gamma++ processes, constructs multidimensional versions, calibrates parameters with real data.
result Closed formulas and numerical methods for evaluating exchange options.
There are several (mathematical) reasons why Dupire's formula fails in the non-diffusion setting. And yet, in practice, ad-hoc preconditioning of the option data works reasonably well. In this note we attempt to explain why. In particular, we propose a regularization procedure of the option data so that Dupire's local …
The paper analyzes Reliability Options in electricity markets, deriving pricing formulas and simulating real market scenarios.
problem Determining the value of Reliability Options in electricity markets under various price regimes.
method The paper derives closed-form pricing formulae and simulates real market scenarios using data from the Italian power market. Sensitivity analyses are performed to highlight the impact of different parameters.
result The value of Reliability Options is influenced by the level and volatility of power and strike prices, mean reversion speeds, and correlation coefficients.
Virtual reality brings non-Euclidean geometry to life.
problem Understanding non-Euclidean geometry is challenging.
method Interactive visualizations in virtual reality.
result Users can experience non-Euclidean geometry firsthand.
The challenge to fruitfully merge state-of-the-art techniques from mathematical finance and numerical analysis has inspired researchers to develop fast deterministic option pricing methods. As a result, highly efficient algorithms to compute option prices in Lévy models by solving partial integro differential equations…
Book provides detailed trading strategies for various asset classes.
problem None explicitly stated, but addresses the need for trading strategies.
method Detailed descriptions and mathematical formulas for over 150 trading strategies.
result Provides comprehensive strategies for multiple asset classes.
We prove Feynman-Kac formulas for solutions to elliptic and parabolic boundary value and obstacle problems associated with a general Markov diffusion process. Our diffusion model covers several popular stochastic volatility models, such as the Heston model, the CEV model and the SABR model, which are widely used as ass…
We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's pri…
Prior design is one of the most important problems in both statistics and machine learning. The cross validation (CV) and the widely applicable information criterion (WAIC) are predictive measures of the Bayesian estimation, however, it has been difficult to apply them to find the optimal prior because their mathematic…
Unified formula for arbitrary liquidity operations in weighted AMMs
problem Decentralized resource allocation in intelligent transportation systems
method Weighted invariant adapted from Balancer-type AMMs
result Unified formula for four resource allocation operations