New integration theory on topological spaces, including fractals.
problem Developing a universal integration theory for arbitrary topological spaces.
method Introducing a new integration framework using unital magma valued functions and measures.
result Integration, differentiation, and orientation defined for arbitrary topological spaces.
Two new versions of a master formula in portfolio theory proven using pathwise Itô calculus.
problem Proving strictly pathwise versions of a master formula in portfolio theory.
method Pathwise Itô calculus, Föllmer's pathwise Itô calculus, Dupire's functional pathwise Itô calculus, Cont & Fournié's functional pathwise Itô calculus.
result Two new versions of the master formula in stochastic portfolio theory proven.
Dupire's functional Itô calculus provides an alternative approach to the classical Malliavin calculus for the computation of sensitivities, also called Greeks, of path-dependent derivatives prices. In this paper, we introduce a measure of path-dependence of functionals within the functional Itô calculus framework. Name…
Extends calculus of variations to generalized functions.
problem Extending calculus of variations to generalized functions.
method Category of generalized smooth functions, proving connections and conditions.
result Full connections between extremals and Euler-Lagrange equations, necessary conditions for minimizers.
Functional calculus results lead to smooth metric dependence on Riemannian geometry.
problem Analyzing smoothness of metric dependence in Riemannian geometry.
method Functional calculus and real analyticity of fractional Laplacians.
result Fractional Laplacians depend real analytically on metrics.
Derives an explicit formula for optimal portfolios in financial markets.
problem Optimal investment problem in complete financial markets driven by Wiener process.
method Functional Itô calculus approach, relying only on integrability condition.
result Derives an explicit formula for the optimal portfolio process.
Extends Euler calculus to continuous integrands using curvature.
problem Limitations of Euler calculus with simple functions.
method Integrates with respect to Gaussian curvature within O-minimal theories.
result Satisfies a Fubini theorem and extends to a functor.
Teaches matrix calculus for machine learning and optimization.
problem Computing derivatives of functions involving matrices.
method Extends differential calculus to vector spaces, focusing on practical applications in machine learning.
result Introduction of adjoint and reverse-mode differentiation for efficient computation.
Survey article on solving geometric problems with calculus and harmonic analysis.
problem Solving geometric problems using functional calculus and harmonic analysis.
method Combining methods from functional calculus and real-variable harmonic analysis.
result Recent geometric problems have been resolved by these methods.
Paper connects differential geometry with geometric calculus.
problem None explicitly stated in the abstract.
method Introduces a more general Laplacian for multivector-valued functions on manifolds.
result Formulated a higher codimensional analog of Jacobi's field equation.
Characterizes polyhomogeneous symbols and applies to Heisenberg calculus.
problem Understanding polyhomogeneous symbols and their applications.
method Simple characterisation and generalization of A.~Connes' tangent groupoid.
result Heisenberg calculus on contact manifolds coincides with groupoid calculus.
Study gradient bounds for Kolmogorov type diffusions using coupling and Γ-calculus.
problem Gradient bounds for Kolmogorov type diffusions.
method Coupling techniques and Γ-calculus.
result Advantages and drawbacks of each method discussed.
Study pseudo-differential operators on compact Lie groups using symbols and functional calculus.
problem Analytical index of pseudo-differential operators on compact Lie groups.
method Use operator-valued symbols and McKean-Singer index formula with operator-valued functional calculus.
result Developed tools for calculating the index of pseudo-differential operators.
Global calculus for manifolds with boundary, solving evolution problems.
problem Global solvability of evolution problems on manifolds with boundary.
method Established global functional calculus and Gårding inequality for pseudo-differential operators without local coordinates.
result Global solvability for a class of evolution problems.
Derives new orthogonal coordinates for evolving surfaces and curves.
problem Accounting for geometric effects in boundary layer asymptotics.
method Elementary derivation of orthogonal signed-distance coordinates.
result Provides vector calculus identities for these coordinates.
This paper is concerned with pseudodifferential calculus on manifolds with fibred corners. Following work of Connes, Monthubert, Skandalis and Androulidakis, we associate to every manifold with fibred corners a longitudinally smooth groupoid which algebraic and differential structure is explicitely described. This grou…
The study characterizes complex structures using calculus of variations.
problem Variational characterization of complex structures.
method Calculus of variations for real vector bundle valued differential forms.
result Obtains variational characterization of complex structures.
We extend projectively equivariant quantization and symbol calculus to symbols of pseudo-differential operators. An explicit expression in terms of hypergeometric functions with noncommutative arguments is given. Some examples are worked out, one of them yielding a quantum length element on S3.
Differential Calculus is a staple of the college mathematics major's diet. Eventually one becomes tired of the same routine, and wishes for a more diverse meal. The college math major may seek to generalize applications of the derivative that involve functions of more than one variable, and thus enjoy a course on Multi…
New calculus solves boundary value problems for elliptic operators.
problem Boundary value problems for 0-elliptic operators.
method Developed a new calculus called symbolic 0-calculus to handle boundary value problems.
result Construct left and right parametrices for 0-elliptic operators with boundary conditions.
Study uses Malliavin calculus to find exact option pricing under stochastic volatility.
problem Exact pricing of European options in markets with stochastic volatility.
method Applying Malliavin calculus to models defined by Ornstein-Uhlenbeck or Cox-Ingersoll-Ross processes.
result Established the density function of the volatility average over time to maturity.
Study computes option sensitivities using Malliavin calculus for hybrid stochastic models.
problem Computing option sensitivities (Greeks) under hybrid stochastic volatility and interest rate models.
method Integrates Malliavin calculus for Delta, Vega, and Rho computation; extends to non-differentiable payoffs.
result Malliavin calculus enables effective numerical implementations for various option types.
The calculus of variations for lagrangians which are not functions on the tangent bundle, but sections certain affine bundles is developed. We follow a general approach to variational principles which admits boundary terms of variations.
Derives FPDE for equity-linked insurance pricing.
problem Calculating prices for insurance policies with complex payment histories.
method Variational techniques in functional Itô calculus.
result Derives a functional partial differential equation.
Extends calculus to topological manifolds using generalized functions.
problem Proving the existence of non-singular generalized tangent vector fields on spheres.
method Develops a theory of generalized functions and applies it to continuous maps between topological spaces.
result Shows coherence between non-existence of smooth vector fields on spheres and existence of generalized ones.
Spectral calculus simplifies manifold learning with eigenfunctions.
problem Learning manifold structures from data.
method Reformulate exterior calculus using Laplacian eigenfunctions.
result Spectral convergence to true exterior calculus.
We introduce a method for evaluating integrals in geometric calculus without introducing coordinates, based on using the fundamental theorem of calculus repeatedly and cutting the resulting manifolds so as to create a boundary and allow for the existence of an antiderivative at each step. The method is a direct general…
In this paper, we extend the DC Calculus introduced by Perelman on finite dimensional Alexandrov spaces with curvature bounded below. Among other things, our results allow us to define the Hessian and the Laplacian of DC functions (including distance functions as a particular instance) as a measure-valued tensor and a …
AP-Calculus offers a new framework for causal inference in Bayesian networks.
problem Causal inference in Bayesian networks with complex architectures.
method Introduces Attribution Projection Calculus (AP-Calculus) to determine causal relationships.
result Proves that for each label, exactly one intermediate node acts as a deconfounder.
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.
Develops a new theory of loss functions for statistical machine learning.
problem Evaluation of solutions in binary and multiclass classification problems.
method Defines loss functions as subgradients of support functions of convex sets, enabling a calculus of losses.
result Provides a novel perspective on losses and develops a calculus that interpolates between different losses.
Optimal regularity for 2D stationary varifolds proved.
problem Regularity of parametrized integer stationary varifolds in 2D.
method Established optimal regularity result for stationary varifolds.
result Smooth minimal branched immersion and constant multiplicity function.
A definition for elliptical tempered stable distribution, based on the characteristic function, have been explained which involve a unique spectral measure. This definition provides a framework for creating a connection between infinite divisible distribution, and particularly elliptical tempered stable distribution, w…
A new method in finance without probabilities or integrals.
problem Creating a model-free approach to continuous-time finance.
method Pathwise approach using causal functional calculus and transition principle of Isaacs.
result A fully non-linear path-dependent equation characterizes optimal solutions.
The paper provides an efficient method to price path-dependent derivatives using multiscale stochastic volatility models.
problem Pricing path-dependent derivatives under multiscale stochastic volatility models.
method Derives a Malliavin representation for the first-order approximation of the price of path-dependent derivatives.
result An efficient Monte Carlo approximation for pricing path-dependent derivatives is derived.
We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold M. We give several applications of this theory, concerning: 1) differentiability and geometrical properties of the distance function to a closed subset C of M; 2) solvability and implicit func…
Abstract: Tangent categories get a Cartan calculus with scalar multiplication by a commutative ring.
problem Constructing a Cartan calculus in tangent categories.
method Define scalar multiplication by a commutative ring object R to equip tangent bundles with R-module structure. result Every object in tangent categories carries a Cartan calculus of Lie-Rinehart forms.
The extremely useful method of Malliavin calculus has not yet gained adequate popularity because of the complicated analytic apparatus of this method. The author attempts here to propose a simplified algebraic formalism similar to Malliavin calculus, but based on the notion of creation-annihilation operators instead of…
Optimizes reinsurance and investment strategies to minimize ruin probability.
problem Optimizing reinsurance and investment strategies to minimize ruin probability.
method Stochastic projected gradient method based on Malliavin calculus.
result Effectiveness of the proposed method demonstrated through numerical experiments.
Study proves Lp properties for Hodge-Dirac operator on curved spaces.
problem Analyzing Lp-behavior of Hodge-Dirac operator on curved manifolds. method Proves R-bisectoriality and boundedness of H∞-functional calculus in Lp. result Establishes Lp-properties for Hodge-Dirac operator on manifolds with non-negative Ricci curvature. 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.
A theorem connects integral of second-order derivatives to function rise.
problem Understanding the integral of second-order derivatives over regions.
method Proves integral proportional to function rise over specified regions.
result Integral of second-order derivatives equals rise in function value.
Develops calculus for conformal hypersurfaces and new Willmore energy functionals.
problem Invariant theory for conformal hypersurfaces.
method Solving singular Yamabe problem, developing calculus of differential operators, computing asymptotics.
result New higher Willmore energy functionals for embedded surfaces.
Study Brownian motion on Grassmann manifold using matrix stochastic calculus.
problem Understanding Brownian motion on non-compact Grassmann manifold.
method Realize Brownian motion as matrix diffusion process, use matrix stochastic calculus, and hyperbolic Stiefel fibration.
result Connection to generalized Maass Laplacian of complex hyperbolic space.
In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects…
Differential forms on the Fréchet manifold F(S,M) of smooth functions on a compact k-dimensional manifold S can be obtained in a natural way from pairs of differential forms on M and S by the hat pairing. Special cases are the transgression map associating (p-k)-forms on F(S,M) to p-forms on M (hat pairing with a const…
Operator calculus for population-based optimization provides a unified framework for analyzing convergence of various methods.
problem Convergence analysis of population-based optimization methods
method Introduce an operator calculus for describing composite mean-field algorithms as compositions of elementary operators acting on probability measures.
result Establish a modular Lyapunov principle for certifying exponential decay of state-space Lyapunov function and search errors.
New calculus on spacetimes for nonlinear differential equations.
problem Nonlinear differential equations on metric measure spacetimes.
method Introduces maximal weak subslope and variational calculus.
result Establishes a comparison theorem for nonlinear p-d'Alembertian.