Study disproves a generalized numerical criterion for certain pairs.
problem Generalized numerical criterion for pairs
method Provided counterexamples
result Negative answer to the generalized numerical criterion problem
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48 results for “numerical example”
Twelve numerical methods for Poisson geometry concepts.
problem Computing and evaluating Poisson geometry concepts.
method Twelve numerical methods for various Poisson geometry operations.
result Experimental verifications of methods in dimensions two and three.
Solves portfolio optimization with costs using numerical methods.
problem Dynamic portfolio optimization with transaction costs and constraints.
method Numerical dynamic programming techniques.
result Problems can now be solved tractably.
We exhibit examples of projective varieties with degenerate Gauss mappings and determine numerical invariants of such varieties. Our examples provide counter-examples to an asserted structure theorem of Griffiths and Harris (Ann. Sci. ENS 1979).
The paper develops Hawkes-based models for LOB and applies them to European, spread, and basket option pricing.
problem Developing accurate models for pricing options in the context of limit order books (LOB).
method Introduces multivariate Hawkes processes and their limit theorems, applies to European, spread, and basket options.
result Hawkes-based models provide more market forecast information than classical models.
Efficient numerical method for time-fractional Black-Scholes model.
problem Solving time-fractional Black-Scholes equations for European options.
method Crank-Nicolson discretization for time, exponential B-spline for space.
result The proposed method is unconditionally stable and superior to existing approaches.
A new numerical scheme approximates nonlinear filtering densities for noisy and partial measurements.
problem Approximating nonlinear filtering densities for noisy and partial measurements.
method Deep splitting scheme applied to the Fokker--Planck equation followed by Bayes' formula.
result Convergence rate established for the numerical scheme under parabolic Hörmander condition.
New numerical method for quantile hedging in imperfect markets.
problem Quantile hedging in non-linear markets with imperfections.
method Piecewise Constant Policy Timestepping (PCPT) coupled with monotone finite difference approximation.
result Convergence of the proposed numerical scheme proved using BSDE arguments.
Numerical groupsmath.GR
Groups of matrices with integer-like entries are studied.
problem Characterizing groups of matrices with algebraic integer entries.
method Analyzing traces and subgroups of matrices in number fields.
result Irreducible or completely reducible subgroups with algebraic integer traces are numerical.
Paper approximates fractional harmonic maps with numerical methods.
problem Approximating fractional harmonic maps with constraints and nonlocality.
method Weak compactness results and numerical methods for various PDEs.
result Convergence of numerical approximations for fractional harmonic maps.
A contour integral method recently proposed by Weideman [IMA J. Numer. Anal., to appear] for integrating semi-discrete advection-diffusion PDEs, is extended for application to some of the important equations of mathematical finance. Using estimates for the numerical range of the spatial operator, optimal contour parame…
The paper connects geometric structures to mechanical systems and introduces new numerical methods.
problem Analyzing and simulating systems with constraints.
method Introduces Dirac integrators based on generalized geometry.
result Dirac integrators can conserve constraints and provide new insights.
Smooth classifying spacesmath.DG
Smooth classifying spaces for groups defined using diffeological spaces.
problem Classifying smooth principal bundles for a smooth group .
method Developed the theory of smooth principal bundles using diffeological spaces, defining -numerable bundles and proving classification results.
result Smooth structures on Milnor's spaces and classify all -numerable principal bundles over any diffeological space.
New method for elastic curve and surface matching.
problem Elastic matching of unparametrized curves and surfaces.
method Combines square root normal fields and varifold fidelity metrics.
result Numerical examples demonstrate the approach's effectiveness.
Nonlinear PDEs risen when solving some optimization problems in finance, and their solutionsq-fin.CP
Nonlinear PDEs in finance optimization problems solved numerically and analytically.
problem Nonlinear PDEs in finance optimization problems.
method Numerical and analytical solutions.
result Solutions to specific PDEs in finance.
Calculates Laplacian spectra on Calabi-Yau hypersurfaces.
problem Computing the spectrum of the Laplacian on complex manifolds.
method Numerical computation of eigenvalues and eigenmodes for line bundles.
result Agreement with exact results for and a torus, first numerical results for Fermat quintic.
These lecture notes provide some introduction to the 3+1 formalism of general relativity, which is the foundation of most modern numerical relativity. The text is rather self-contained, with detailed calculations and numerous examples. Contents: 1. Introduction, 2. Geometry of hypersurfaces, 3. Geometry of foliations, …
We recall the theory of linear discrete Riemann surfaces and show how to use it in order to interpret a surface embedded in R^3 as a discrete Riemann surface and compute its basis of holomorphic forms on it. We present numerical examples, recovering known results to test the numerics and giving the yet unknown period m…
We introduce a new class of local volatility models. Within this framework, we obtain expressions for both (i) the price of any European option and (ii) the induced implied volatility smile. As an illustration of our framework, we perform specific pricing and implied volatility computations for a CEV-like example. Nume…
We apply results of Malliavin-Thalmaier-Watanabe for strong and weak Taylor expansions of solutions of perturbed stochastic differential equations (SDEs). In particular, we work out weight expressions for the Taylor coefficients of the expansion. The results are applied to LIBOR market models in order to deal with the …
Study existence of harmonic 1-forms on Calabi-Yau manifolds.
problem Tackles existence of harmonic 1-forms on Calabi-Yau manifolds.
method Uses neural networks to approximate metrics and harmonic 1-forms.
result Suggests existence of harmonic 1-forms on some Calabi-Yau manifolds.
Path Diffusion, Part Iq-fin.MF
This paper investigates the position (state) distribution of the single step binomial (multi-nomial) process on a discrete state / time grid under the assumption that the velocity process rather than the state process is Markovian. In this model the particle follows a simple multi-step process in velocity space which a…
Estimates domain truncation error for option pricing PDEs.
problem Estimating error in option pricing models with domain truncation.
method Derives an estimate of domain truncation error for a multidimensional PDE system.
result Proposes a sharper error estimate for option pricing models.
Optimal Insurance to Maximize Exponential Utility when Premium is Computed by a Convex Functionalq-fin.MF
Optimal insurance policy for exponential utility maximization with convex premium calculation.
problem Maximizing terminal wealth utility with exponential utility function and convex premium formula.
method Necessary condition for optimal indemnity, numerical algorithm to compute it, convergence proof.
result Numerical algorithm converges to unique optimal indemnity.
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…
High order splitting schemes with complex timesteps and their application in mathematical financemath.NA
High order splitting schemes with complex timesteps are applied to Kolmogorov backward equations stemming from stochastic differential equations in Stratonovich form. In the setting of weighted spaces, the necessary analyticity of the split semigroups can be easily proved. A numerical example from interest rate theory,…
The paper explores risk-minimization for exponential additive models, providing mathematical expressions and numerical examples.
problem Risk-minimization in incomplete markets for exponential additive models.
method Derive explicit mathematical expressions for local risk-minimization strategies in exponential additive models.
result Provide necessary conditions for deriving expressions and confirm integrability conditions for specific models.
We provide a bound for the error committed when using a Fourier method to price European options when the underlying follows an exponential \levy dynamic. The price of the option is described by a partial integro-differential equation (PIDE). Applying a Fourier transformation to the PIDE yields an ordinary differential…
A new scheme for FBSDEs simplifies computation without Monte Carlo.
problem Numerical solution for decoupled FBSDEs with reduced complexity.
method Recursive marginal quantization for fully quantization-based scheme.
result Effective numerical procedure for financial applications.
Gradient flow for Willmore energy defined in Sobolev space.
problem Defining -gradient flow for Willmore energy in Sobolev class.
method Introduced -gradient flow for Willmore energy in space, provided and . Discussed existence and usability of projected flow.
result Existence and usability of -gradient flow for optimization.
New numerical methods for evolving curves on curved spaces.
problem Evolve curves on Riemannian manifolds efficiently and accurately.
method Variational approximations and numerical schemes for curvature flow, curve diffusion, and elastic flow.
result Effective numerical schemes for geometric evolution equations on Riemannian manifolds.
Lectures on Moebius-Lie geometry and its extension, including new geometric ensembles.
problem Classical Moebius-Lie geometry and its extension to ensembles of cycles.
method Reduces conformally invariant geometric relations to linear equations with a fixed quadratic relation.
result Efficient method implemented as a C++ library for numeric and symbolic data in arbitrary dimensions.
This paper describes a consistent and arbitrage-free pricing methodology for bespoke CDO tranches. The proposed method is a multi-factor extension to the (Li 2009) model, and it is free of the known flaws in the current standard pricing method of base correlation mapping. This method assigns a distinct market factor to…
New method integrates latent variables for Bayesian Optimization of materials with both qualitative and quantitative factors.
problem Bayesian Optimization for materials design with mixed qualitative and quantitative variables.
method Integrates latent variables for mixed-variable Gaussian process modeling within the Bayesian Optimization framework.
result LVGP provides superior modeling accuracy compared to existing methods for mixed-variable problems.
New deep architectures inspired by numerical differential equations improve performance.
problem Designing effective deep neural networks.
method Interpreting deep networks as numerical discretizations of differential equations.
result LM-ResNet and LM-ResNeXt achieve higher accuracy with less parameters.
This paper explores probabilistic numerical methods for integrating statistical computations.
problem Handling numerical error as epistemic uncertainty in statistical computation.
method Probabilistic integrators that model numerical error as a distribution.
result Probabilistic integrators can achieve posterior contraction rates similar to Monte Carlo methods.
The paper applies deep learning to solve complex control problems.
problem Solving stochastic control problems on finite horizons.
method Deep neural networks algorithms for control learning.
result Performance of algorithms on various control problems demonstrated.
New algorithms recover differential equations from short bursts of data.
problem Locally recover unknown governing differential equations from measurement data.
method Approximate governing equations using standard basis functions and short bursts of trajectory data.
result Effective numerical algorithms recover accurate governing equations from short bursts of data.
A new method codes nominal data as complex numbers for better classification.
problem Losing information in nominal data coding for effective classification.
method Assigning a rank as a complex number to nominal data for classification.
result Classification with coded nominal data is more effective than with only numerical data.
The paper compares numerical schemes for nonholonomic systems using retraction maps.
problem Optimal control of nonholonomic systems with numerical approximations.
method Retraction maps used as seed for geometric integrators of Hamilton equations.
result Performance comparison of symplectic and non-symplectic integrators.
We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite dimensional Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric and on a family o…
The paper develops option pricing methods for bilateral Gamma stock models.
problem Developing accurate option pricing measures for bilateral Gamma stock models.
method Incorporates various mathematical techniques including Esscher transforms, minimal entropy martingale measures, and -optimal martingale measures.
result Illustrates the theory with a numerical example, providing practical application of the methods.
New method uses reinforced regression for solving optimal stopping problems.
problem Solving optimal stopping problems in mathematical finance.
method Reinforced regression based on previously estimated continuation values.
result Illustrated by a numerical example from mathematical finance.
This paper performs the numerical analysis and the computation of a Spread option in a market with imperfect liquidity. The number of shares traded in the stock market has a direct impact on the stock's price. Thus, we consider a full-feedback model in which price impact is fully incorporated into the model. The price …
New high-order compact scheme improves basket option pricing accuracy.
problem Improving accuracy in pricing European Put options on a basket of assets.
method Developed a second-order accurate in time and fourth-order accurate in space high-order compact scheme.
result Standard second-order schemes are significantly outperformed by the new scheme.
The paper surveys mathematical results on filtration enlargement with financial examples.
problem Mathematical finance applications of filtration enlargement theory.
method Exhaustive survey and interpretation of key results from literature.
result Provides a compendium of known mathematical results for mathematical finance researchers.
This paper develops numerical methods for finding optimal dividend pay-out and reinsurance policies. A generalized singular control formulation of surplus and discounted payoff function are introduced, where the surplus is modeled by a regime-switching process subject to both regular and singular controls. To approxima…
Efficient method for lookback option pricing under Markov models.
problem Pricing lookback options under Markov models.
method Model-free representations combined with numerical quadrature and Markov chain approximation.
result Efficient method applicable to various Markov models.