It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwe…
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A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
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48 results for “underlying equations”
We propose a new two-component geodesic equation with the unusual property that the underlying space has constant positive curvature. In the special case of one space dimension, the equation reduces to the two-component Hunter-Saxton equation.
Classifies solutions to vacuum weighted Einstein equations on pr-waves.
problem Classifying solutions to vacuum weighted Einstein field equations on pr-waves.
method Classifying solutions using smooth metric measure spacetimes of dimension 4.
result Provides examples of solutions with special geometric properties.
This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler-L…
FDNet learns PDEs from data with fast predictions.
problem Discovering complex systems behavior from data.
method Finite difference neural networks (FDNet) to learn PDEs from trajectory data.
result FDNet predicts future behavior with few trainable parameters.
Two proxy methods for causal identification are compared.
problem Identifying causal effects in the presence of unmeasured variables.
method Bridge equation methods vs. array decomposition methods.
result Model restrictions and implications of assumptions differ between methods.
In this paper we will develop a methodology for obtaining pricing expressions for financial instruments whose underlying asset can be described through a simple continuous-time random walk (CTRW) market model. Our approach is very natural to the issue because it is based in the use of renewal equations, and therefore i…
We derive a forward partial integro-differential equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a -possibly discontinuous- semimartingale. A uniqueness theorem is given for the solutions of this equation. This result generalizes Dupire…
Method learns latent SDEs from high-dimensional time series.
problem Learning latent stochastic differential equations from time series data.
method Self-supervised learning with variational autoencoders and Euler-Maruyama approximation.
result Can recover SDE coefficients and latent variables up to isometry with infinite data.
Solves a generalized Monge-Ampère equation on Kähler surfaces, proving a conjecture.
problem Existence and uniqueness of solutions to a generalized Monge-Ampère equation.
method Analyzes the equation's dependence on almost Kähler structure and proves Donaldson's conjecture.
result Proves Donaldson's conjecture for tamed almost complex 4-manifolds.
The paper explores the geometry of the Spence-Kummer trilogarithm equation and its Galois analogue.
problem Investigating the geometry and functional equation of the Spence-Kummer trilogarithm.
method Using algebraic relations between polylogarithm generating series and path systems, along with tensor and homotopy criteria for functional equations.
result Derives a precise form of the Spence-Kummer equation and its Galois analogue.
Geometric characterization and classification of Bäcklund transformations of sine-Gordon typemath.DG
We begin by considering several properties commonly (but not universally) possessed by Bäcklund transformations between hyperbolic Monge-Ampère equations: wavelike nature of the underlying equations, preservation of independent variables, quasilinearity of the transformation, and autonomy of the transformation. We show…
A master equation approach to option pricingcond-mat.stat-mech
A master equation approach to the numerical solution of option pricing models is developed. The basic idea of the approach is to consider the Black--Scholes equation as the macroscopic equation of an underlying mesoscopic stochastic option price variable. The dynamics of the latter is constructed and formulated in term…
ERFit identifies dynamic equations from data with minimal supervision.
problem Data-driven sparse system identification in science and engineering.
method Entropic Regression method.
result ERFit package simplifies sparse system identification for various applications.
Novel framework discovers SPDEs from limited data.
problem Discovering SPDEs from limited data.
method Combines stochastic calculus, variational Bayes, and sparse learning.
result Accurately identifies SPDEs from limited data.
We consider the mean-variance hedging problem under partial information in the case where the flow of observable events does not contain the full information on the underlying asset price process. We introduce a martingale equation of a new type and characterize the optimal strategy in terms of the solution of this equ…
We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of the underlying PDE numerically. The evolution operator of the PDE, defined in i…
We study the limiting behaviour of solutions to abelian vortex equations when the volume of the underlying Riemann surface grows to infinity. We prove that the solutions converge smoothly away from finitely many points. The proof relies on a priori estimates for functions satisfying generalised Kazdan-Warner equations.…
Conformal Fields and the Structure of the Space of Solutions of the Einstein Constraint Equationsgr-qc
The drift method, introduced by the second author, provides a new formulation of the Einstein constraint equations, either in vacuum or with matter fields. The natural of the geometry underlying this method compensates for its slightly greater analytic complexity over, say, the conformal or conformal thin sandwich meth…
Given an irreducible unitary representation of a cocompact lattice of SL(2,C), we explicitly write down a solution of the Strominger system of equations. These solutions satisfy the equation of motion, and the underlying holomorphic vector bundles are stable.
The paper studies the First Order BSPDEs (Backward Stochastic Partial Differential Equations) suggested earlier for a case of multidimensional state domain with a boundary. These equations represent analogs of Hamilton-Jacobi-Bellman equations and allow to construct the value function for stochastic optimal control pro…
This paper surveys options pricing under arithmetic Brownian motion and derives formulas for various types of options.
problem The use of arithmetic Brownian motion in finance is not widely adopted.
method Risk-neutral valuation and derivation of formulas for European options under three types of underlying assets.
result Derivation of formulas for European options and partial differential equations for American options.
Convergence of fundamental solutions of linear parabolic equations under Cheeger-Gromov convergencemath.DG
In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger-Gromov convergence of the underlying manifolds and the uniform -bound of the solutions. We also prove a local integral estimate of fundamental solutions.
New boundary condition for Black-Scholes equations in strict local martingale models.
problem Computing prices of European options with underlying asset as a strict local martingale.
method Numerical procedure using finite difference methods with a new boundary condition at infinity.
result The minimal solution, satisfying a discrete maximum principle, is the correct derivative price.
TNet combines DL with physics models to solve inverse problems efficiently.
problem Solving inverse problems with limited data and physics constraints.
method Model-constrained deep learning approach using TNet.
result TNet solutions are as accurate as traditional methods but faster.
Recent works have derived non-asymptotic upper bounds for convergence of underdamped Langevin MCMC. We revisit these bound and consider introducing scaling terms in the underlying underdamped Langevin equation. In particular, we provide conditions under which an appropriate scaling allows to improve the error bounds in…
We prove that the geodesic equations of all Sobolev metrics of fractional order one and higher on spaces of diffeomorphisms and, more generally, immersions are locally well posed. This result builds on the recently established real analytic dependence of fractional Laplacians on the underlying Riemannian metric. It ext…
Extends Zeitlin's model to 3-D axisymmetric Euler equations.
problem Preserving geometric structure in 3-D Euler equations.
method Axisymmetric discretization of 3-D Euler equations on the 3-sphere.
result First discretization of 3-D Euler equations preserving geometric structure.
New method uses entropy to generate multiple plausible causal maps.
problem Learning causal relationships from noisy data can lead to artifacts in DAGs.
method Entropy-based inference to generate an ensemble of plausible causal graphs.
result Multiple causal maps consistent with underlying data variability.
In this paper we introduce invariants of semi-free Hamiltonian actions of $S\sp 1$ on compact symplectic manifolds (which satisfy some technical conditions related to positivity) using the space of solutions to certain gauge theoretical equations. These equations generalize at the same time the vortex equations and the…
We study (backward) stochastic differential equations with noise coming from a finite state Markov chain. We show that, for the solutions of these equations to be `Markovian', in the sense that they are deterministic functions of the state of the underlying chain, the integrand must be of a specific form. This allows u…
Stochastic Normalizing Flowsstat.ML
Stochastic normalizing flows use SDEs for efficient training and sampling.
problem Efficient maximum likelihood estimation and variational inference.
method Continuous normalizing flows extended with stochastic differential equations (SDEs) and rough path theory.
result Stochastic normalizing flows enable efficient training and sampling from complex distributions.
In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Her…
We provide a moment map interpretation for the coupled Kähler-Einstein equations introduced by Hultgren and Witt Nyström, and in the process introduce a more general system of equations, which we call coupled cscK equations. A differentio-geometric formulation of the corresponding Futaki invariant is obtained and a not…
It is pointed out that despite of the non-linearity of the underlying equations, there do exist rather general methods that allow to generate new minimal surfaces from known ones.
IGNN captures long-range graph dependencies using fixed-point equations.
problem Limited GNN ability to capture long-range graph dependencies.
method Fixed-point equilibrium equations involving implicitly defined state vectors, leveraging Perron-Frobenius theory and projected gradient descent.
result IGNN consistently captures long-range dependencies and outperforms state-of-the-art GNNs.
We give multiplicity results for the solutions of a nonlinear elliptic equation, with an asymmetric double well potential of Van der Waals-Allen--Cahn--Hilliard type, satisfying a linear volume constraint, on a bounded Lipschitz domain $Ω\subset\mathds R^N$. The number of solutions is estimated in terms of topological …
In this paper we analyze American style of floating strike Asian call options belonging to the class of financial derivatives whose payoff diagram depends not only on the underlying asset price but also on the path average of underlying asset prices over some predetermined time interval. The mathematical model for the …
We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we develop such connections for the case of two spatial dimensions, and demonstrate tha…
Solves geometric problems using fully nonlinear equations and Morse theory.
problem Geometric problems, specifically Loewner-Nirenberg and Yamabe problems.
method Investigates structure of fully nonlinear equations and applies Morse theory techniques.
result Constructs admissible metrics under weak conditions and demonstrates topological obstructions.
Study uses machine learning to predict predator-prey dynamics without prior knowledge.
problem Predicting predator-prey interactions without prior knowledge of the system.
method Applied Neural Ordinary Differential Equations (Neural ODEs) and Universal Differential Equations (UDEs) to the Lotka-Volterra model.
result UDEs outperform Neural ODEs in predicting predator-prey dynamics, especially in noisy data.
Post-groupoids help solve Yang-Baxter equation using quivers.
problem Solving the Yang-Baxter equation using algebraic structures.
method Introducing post-groupoids and showing their connection to quivers.
result Post-groupoids provide solutions to the Yang-Baxter equation.
New quantum algorithm simplifies complex financial derivatives pricing.
problem Complex financial derivatives pricing with high dimensionality.
method Quantum-inspired variational algorithms combined with neural-network quantum states.
result Simplified pricing of European options with many correlated assets.
Study Liouville equation on Riemannian surfaces, linking volume growth to classification results.
problem Classifying solutions and manifolds of the Liouville equation on Riemannian surfaces.
method Analyzing the Liouville equation on Riemannian surfaces with non-negative Ricci curvature, considering asymptotic volume growth.
result Established classification results for solutions and manifolds, revealing a connection between volume growth and classification.
Solutions of the equation of a spinorial Yamabe-type problem on manifolds of bounded geometrymath.DG
We consider a spinorial Yamabe-type problem on open manifolds of bounded geometry. The aim is to study the existence of solutions to the associated Euler-Lagrange-equation. We show that under suitable assumptions such a solution exists. As an application, we prove that existence of a solution implies the conformal Hija…
In this paper we provide a variational derivation of the Euler-Poincaré equations for systems subjected to external forces using an adaptation of the techniques introduced by Galley and others. Moreover, we study in detail the underlying geometry which is related to the notion of Poisson groupoid. Finally, we apply the…
We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second fundamental form of its boundary. The main result (Theorem 1.1) includes a new (and opt…
In this article, we consider a gauge-theoretic equation on compact symplectic 6-manifolds, which forms an elliptic system after gauge fixing. This can be thought of as a higher-dimensional analogue of the Seiberg-Witten equation. By using the virtual neighbourhood method by Ruan, we define an integer-valued invariant, …