The paper solves a complex option pricing model using finite elements.
problem Risk-Adjusted Pricing Methodology (RAPM) Black-Scholes model with transaction costs.
method Spatial finite element models based on P1 and/or P2 elements, combined with a Crank-Nicolson-type temporal scheme.
result Results compare favorably with finite difference methods in the literature.
Finite element method applied to Leland's model for option pricing with transaction costs.
problem Option pricing with transaction costs using Leland's model.
method Spatial finite element models based on P1 and/or P2 elements combined with a Crank-Nicolson-type temporal scheme.
result Results compare favorably with finite difference methods in the literature.
We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility jump models, e.g. in Bates model. In such models the option price is determined as the solution of a partial integro-differential equation. The scheme is fourth order accurate in space and second order accurate in ti…
Paper solves convertible bond valuation using finite elements with penalty method.
problem Valuation of convertible bonds under penalty TF model.
method Solves TF system of equations using P1 and P2 finite elements with penalty method.
result Numerical solutions compare favorably with finite difference method.
Variational approximations for curve flows on Riemannian manifolds.
problem Approximating solutions to curvature and elastic flow problems on Riemannian manifolds.
method Variational formulations, finite element approximations, piecewise linear elements, stability analysis.
result Derived schemes can compute rotationally symmetric self-shrinkers and geodesics.
Study uses FEM for HJB in option pricing with borrowing fees, improving accuracy and efficiency.
problem Optimal control problems in financial markets with frictions.
method Finite element method with non-uniform mesh, theta-scheme time integration, Newton-type algorithm.
result Efficient and accurate solution to HJB equation for option pricing with borrowing fees.
New LSTM scheme incorporates prior knowledge and measurement uncertainties.
problem Overfitting and insufficient data for accurate time-dependent solutions.
method Sparse Bayesian training algorithm for automatic connection determination.
result Less prone to overfitting, smaller data set required for satisfying accuracy.
Study finds minimal length networks connecting three points in Heisenberg group.
problem Finding minimal length networks connecting three points in the Heisenberg group.
method Proved existence of minimal horizontal triods, formulated curve shortening flow, used numerical experiments.
result Characterized and deformed minimal horizontal triods into critical points for length functional.
A new method for pricing options with stochastic volatility and jumps.
problem Pricing options under stochastic volatility and jumps.
method Fourth-order compact finite-difference scheme with implicit-explicit Crank-Nicolson framework.
result The method achieves near-fourth-order spatial accuracy and up to two orders of magnitude lower runtime than quadratic finite elements.
We discuss some differential geometry pertaining to continuum mechanics and the route recently taken by D.N. Arnold, R.S. Falk, and R. Winther in deriving new improved finite element schemes in linear elasticity from constructions in projective geometry.
New method approximates anisotropic curve shortening flow.
problem Approximating anisotropic curve shortening flow.
method Weak formulation and finite element approximation.
result Optimal H1-error bound for approximation. We propose a deterministic numerical method for pricing vanilla options under the SABR stochastic volatility model, based on a finite element discretization of the Kolmogorov pricing equations via non-symmetric Dirichlet forms. Our pricing method is valid under mild assumptions on parameter configurations of the proces…
This paper generalizes the Maurer--Pontil framework of finite-dimensional lossy coding schemes to the setting where a high-dimensional random vector is mapped to an element of a compact set of latent representations in a lower-dimensional Euclidean space, and the reconstruction map belongs to a given class of nonlinear…
New boundary treatment improves accuracy for complex PDEs.
problem Order reduction in high-order IMEX schemes for multidimensional PDEs.
method Novel boundary treatment algorithms for Cartesian meshes, treating implicit-explicit stages similarly to interior points.
result Recovery of designed order of convergence by numerical verification.
The signed volume function for polyhedra can be generalized to a mean volume function for volume elements by averaging over the triangulations of the underlying polyhedron. If we consider these up to translation and scaling, the resulting quotient space is diffeomorphic to a sphere. The mean volume function restricted …
New finite element method for complex forms in any dimension.
problem Discretization of complex forms in arbitrary dimensions.
method Finite element discretization of ℓ-form-valued k-forms on triangulations. result Generalizes existing finite element methods for various tensor fields.
The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.
problem Approximating Levi-Civita connection and curvature on 2D manifolds with finite elements.
method Using Regge finite elements, piecewise polynomial symmetric (0,2)-tensor fields, and distributional sense for non-regular tensors.
result Distributional quantities converge to their smooth counterparts under refinement of triangulation.
The paper improves convergence rates of curvature approximations using Regge elements.
problem Improving convergence rates of curvature approximations using Regge elements.
method Investigates the interplay between polynomial degree of curvature lifting and metric tensor degree in Regge finite element space.
result Higher convergence rates are achieved by reducing the polynomial degree of curvature lifting and using linear Regge elements.
Constructs finite element spaces for (p,q)-forms, excluding one subspace.
problem Constructing finite element spaces for (p,q)-forms. method Piecewise polynomial finite element spaces for all natural subspaces of (p,q)-forms, excluding one subspace. result Recovers known finite element spaces and introduces new ones.
Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other discretisation schemes. There are, however, several theoretical approaches to systems…
A recent paper of Arnold, Falk, and Winther [Bull AMS, 47 (2010)] showed that a large class of mixed finite element methods can be formulated naturally on Hilbert complexes, where using a Galerkin-like approach, one solves a variational problem on a finite-dimensional subcomplex. In a seemingly unrelated research direc…
Study SL(2,C) character schemes for finitely generated groups.
problem Characterize SL(2,C) representations of finitely generated groups.
method Define coordinate rings and equations for SL(2,C) character schemes.
result Explicit equations for character schemes of finitely presented groups.
Characterizes periodic elements in Artin-Tits groups via stability conditions.
problem Understanding periodic elements in Artin-Tits groups.
method Dynamical characterization via 2-Calabi-Yau category and stability conditions.
result An element is periodic if and only if it has a fixed point in the stability manifold.
Two elements generate extended mapping class groups of certain surfaces.
problem Generating extended mapping class groups with specific elements.
method Analyzing finite order elements and isotopy classes of homeomorphisms.
result Extended mapping class groups of certain surfaces are generated by two elements of finite order.
A positive integer m will be called a {\it finitistic order} for an element γ of a group Γ if there exist a finite group G and a homomorphism h:Γ→G such that h(γ) has order m in G. It is shown that up to conjugacy, all but finitely many elements of a given finitely generated, torsion-free Kleinian gr…
Develops a new solver for path-dependent PDEs using signature kernels.
problem Solving path-dependent PDEs (PPDEs) efficiently and accurately.
method Uses signature kernels to solve PPDEs by approximating the solution with minimal norm in a reproducing kernel Hilbert space.
result Proves the consistency of the numerical scheme, ensuring convergence to PPDE solutions as the number of collocation points increases.
Let G be a finitely generated group with a given word metric. The asymptotic density of elements in G that have a particular property P is defined to be the limit, as r goes to infinity, of the proportion of elements in the ball of radius r which have the property P. We obtain a formula to compute the asymptotic densit…
Rust library solves complex equations on abstract simplicial complexes.
problem Solving partial differential equations on abstract simplicial complexes.
method Finite Element Exterior Calculus, intrinsic Riemannian metric, first-order Whitney basis functions.
result Verification through convergence studies on elliptic Hodge-Laplace eigenvalue and source problems.
Statistical finite elements use Langevin dynamics to efficiently handle uncertainty quantification.
problem Uncertainty quantification in finite element models with observed data.
method Langevin dynamics, unadjusted Langevin algorithm (ULA), for sampling posterior distributions.
result ULA provides a scalable and efficient method for characterizing the posterior distribution of statFEM models.
If F is a surface with boundary, then a finitely generated subgroup without peripheral elements of G = π_1(F) can be separated from finitely many other elements of G by a finite index subgroup of G corresponding to a finite cover F' with the same number of boundary components as F .
Within the framework of statistical learning theory we analyze in detail the so-called elastic-net regularization scheme proposed by Zou and Hastie for the selection of groups of correlated variables. To investigate on the statistical properties of this scheme and in particular on its consistency properties, we set up …
The paper analyzes finite element methods on manifolds with approximate metrics.
problem Analyzing finite element methods on manifolds with approximate metrics.
method Intrinsic finite element exterior calculus applied to manifolds with Regge metrics.
result Analysis and implementation of a method for computing an approximate Levi-Civita connection form.
Finite quandles with n elements can be represented as n-by-n matrices. We show how to use these matrices to distinguish all isomorphism classes of finite quandles for a given cardinality n, as well as how to compute the automorphism group of each finite quandle. As an application, we classify finite quandles with up to…
Given a finite set of r points in a closed surface of genus g, we consider the torsion elements in the mapping class group of the surface leaving the finite set invariant. We show that the torsion elements generate the mapping class group if and only if (g,r)=(2,5k+4) for some integer k.
Vector fields on schemes have flows if rings are finitely generated.
problem Understanding vector fields and flows on schemes.
method Analyzing vector fields on affine C∞-schemes with finitely generated rings. result Vector fields on affine C∞-schemes with finitely generated rings have flows and are groupoid internal maps. Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension n, arbitrary polynomial degree r, and arbitrary differential form degree k. The study of finite …
The paper constructs discrete Hessian and divdiv complexes on triangulations and proves their cohomology isomorphic to continuous versions.
problem Discrete construction of Hessian and divdiv complexes on triangulations.
method Construction of discrete Hessian and divdiv complexes using finite elements and Dirac measures on triangulations.
result The cohomology of the constructed complexes is isomorphic to the continuous de Rham cohomology.
We evaluate the hedging performance of a high-order compact finite difference scheme from [4] for option pricing in Bates model. We compare the scheme's hedging performance to standard finite difference methods in different examples. We observe that the new scheme outperforms a standard, second-order central finite dif…
Paper develops a finite dimensional approximation scheme for Riemannian manifolds.
problem Integration on Riemannian manifolds.
method New finite dimensional approximation scheme motivated by categorical colimit.
result Establishes a generalization for L1-functionals on Riemannian manifolds. We enhance the biquandle counting invariant using elements of truncated biquandle-labeled Polyak algebras. These finite type enhancements reduce to the finite type enhancements defined by Goussarov, Polyak and Viro for the trivial biquandle of one element and determine (but are not determined by) the biquandle counting…
EPGP surrogate outperforms finite elements in solving wave equations.
problem Benchmarking Gaussian Process surrogates vs. finite elements for wave equation solutions.
method EPGP uses penalized least squares and exponential-polynomial bases; CN-FEM employs Crank--Nicolson time stepping.
result EPGP achieves lower error than CN-FEM under matched degrees-of-freedom.
Extends Nielsen realization to infinite-type surfaces, classifying torsion elements and topological groups.
problem Realizing finite subgroups of mapping class groups on infinite-type surfaces.
method Extending Kerckhoff's result to infinite-type surfaces, using hyperbolic metrics and topological group properties.
result Compact subgroups of mapping class groups are finite, and locally compact subgroups are discrete.
Develops unisolvent weights for Nédélec second family finite elements in 2D.
problem Finding efficient degrees of freedom for Nédélec second family finite elements.
method Uses techniques of homological algebra to obtain degrees of freedom for differential forms.
result Provides a family of unisolvent and minimal physical degrees of freedom for Nédélec second family finite elements.
We construct examples of finite covers of punctured surfaces where the first rational homology is not spanned by lifts of simple closed curves. More generally, for any set O⊂Fn which is contained in the union of finitely many Aut(Fn)-orbits, we construct finite-index normal subgroups of Fn wh…
Study shows Morse elements are common in acylindrically hyperbolic groups.
problem Understanding generic elements in acylindrically hyperbolic groups.
method Analyzing Morse elements and outer automorphisms.
result Morse elements are common in acylindrically hyperbolic groups.
Finite element method approximates scalar curvature in arbitrary dimensions.
problem Approximating scalar curvature using finite elements in arbitrary dimensions.
method Piecewise polynomial interpolants of a smooth Riemannian metric on a triangulated polyhedral domain.
result Finite element interpolants converge to scalar curvature with rate O(hr+1) in H−2(Ω) norm. The study optimizes Gaussian process approximations for finite-rank models.
problem Posterior behavior of finite-rank approximations differs from parent GP priors.
method Locally supported basis expansions with dependent Gaussian coefficients.
result Finite-rank expansions inherit the same posterior contraction rate as parent GP priors.
Constructs CAT(0) actions for certain groups without unipotent elements.
problem Understanding actions of certain groups on CAT(0) spaces.
method Constructs an isometric action of a group on a CAT(0) space.
result Fundamental groups of certain 3-manifolds do not admit faithful finite-dimensional unitary representations.