Derives formula for skew stickiness ratio in asset price and volatility dynamics.
problem Capturing joint dynamics of asset price and volatility.
method Uses Itô-Wentzell and Clark-Ocone formulae to derive representation.
result Derives asymptotics of skew stickiness ratio under stochastic volatility models.
Paper derives a Reilly type integral formula and applies it to inequalities and eigenvalue problems.
problem Developing a new integral formula and its applications in geometric inequalities and eigenvalue problems.
method Derives a Reilly type integral formula associated with the φ-Laplacian and applies it to inequalities and eigenvalue problems. result Obtains Heintze-Karcher and Minkowski type inequalities, and eigenvalue relationships.
Derives new equations for stochastic volatility models.
problem Modeling local-stochastic-volatility models and their derivatives.
method Conditional forward equation, Dupire stochastic PDE, rolling expiry vanilla option SPDE.
result New equations for LSV models and their derivatives.
Derives new equations for volatility models and option pricing.
problem Modeling and pricing options in local-stochastic-volatility models.
method Develops conditional forward equations and Dupire stochastic PDEs.
result Derives new SPDE for vanilla options.
Estimates spectral gap for Brownian motion on sticky-reflecting domains.
problem Estimating spectral gap for Brownian motion on sticky-reflecting domains.
method Interpolation method and novel applications of Reilly formula.
result Lower bounds for spectral gap derived for general domains.
Sharp bounds and rigidity theorems for eigenvalues on manifolds.
problem Estimating eigenvalues and characterizing rigidity on manifolds.
method Volume comparison, Escobar-type eigenvalue comparisons, and Reilly formula.
result Sharp bounds and rigidity conditions for eigenvalues on manifolds.
The paper develops a method for stochastic differential equations on manifolds using Schwartz morphisms and diffusion generators.
problem Representing stochastic differential equations on smooth manifolds.
method Using Schwartz morphisms and diffusion generators to construct SDEs on manifolds.
result An extended Ito formula for SDEs on manifolds.
Lower bounds for eigenvalues on manifolds with boundary conditions.
problem Eigenvalue bounds for manifolds with boundary conditions.
method Proving lower bounds for the first non-trivial eigenvalue using Cheeger-type constants.
result Results in the spirit of Cheeger's inequality for manifolds with boundary conditions.
In this paper, we prove some isoperimetric bounds for lower order eigenvalues of the Wentzell-Laplace operator on bounded domains of a Euclidean space or a Hadamard manifold, of the Laplacian on closed hypersurfaces of a Euclidean space or a Hadamard manifold, and of a biharmonic Steklov problem on bounded domains of a…
The one-dimensional SDE with non Lipschitz diffusion coefficient dXt=b(Xt)dt+σXtγdBt, X0=x, γ<1 is widely studied in mathematical finance. Several works have proposed asymptotic analysis of densities and implied volatilities in models involving instances of this equation, based on a careful i…
The paper establishes sharp geometric inequalities for hypersurfaces in warped product manifolds.
problem Geometric inequalities involving three distinct quantities in warped product manifolds.
method Two families of inequalities comparing three geometric quantities in space forms or warped product manifolds.
result Generalizes and extends previous results on Weinstock-type inequalities and Steklov/Wentzell eigenvalues.
Moving boundary problems allow to model systems with phase transition at an inner boundary. Driven by problems in economics and finance, in particular modeling of limit order books, we consider a stochastic and non-linear extension of the classical Stefan-problem in one space dimension, where the paths of the moving in…
A new method uses deep learning to predict rare events in complex systems.
problem Predicting rare and extreme events in non-equilibrium systems.
method A deep learning approach that minimizes the geometrical action.
result The method accurately predicts rare events in various complex systems.
In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem of the weighted La…
Study large deviations for hypoelliptic diffusion on sub-Riemannian manifolds.
problem Large deviations for hypoelliptic diffusion measures on sub-Riemannian manifolds.
method Rough path theory and manifold-valued Malliavin calculus.
result Proved a large deviation principle for pinned hypoelliptic diffusion measures.
Deep reinforcement learning method finds rare events in complex systems.
problem Computing transition pathways in high-dimensional systems.
method Formulated as a cost minimization problem, solved using DDPG with physical properties.
result Efficiently samples and computes globally optimal transition pathways.
We generalize classical large deviations theorems to the setting of complete Riemannian manifolds. We prove the analogue of Mogulskii's theorem for geodesic random walks via a general approach using visocity solutions for Hamilton-Jacobi equations. As a corollary, we also obtain the analogue of Cramér's theorem. The ap…
Paper finds lower bounds for eigenvalues of Bi-drifted Laplacian on smooth metric measure spaces.
problem Eigenvalue problems for Bi-drifted Laplacian on compact manifolds with boundary conditions.
method Obtained lower bounds using specific curvature conditions.
result Lower bounds for the first eigenvalue of Bi-drifted Laplacian.
Study bounds for Brownian motion on manifolds with sticky boundary conditions.
problem Proving geometric bounds for Brownian motion on manifolds with sticky boundary conditions.
method Interpolation involving energy interactions between boundary and interior of the manifold.
result Explicit geometric bounds on Steklov eigenvalues, boundary trace operators, and boundary trace logarithmic Sobolev constants.
New framework embeds generalization in learning dynamics using large deviation theory.
problem Improving generalization and robustness in learning problems.
method Gradient methods from continuous-time perspective with Freidlin-Wentzell theory of large deviations.
result Asymptotic probability estimate for rare events in learning dynamics.
We consider a dynamic market model where buyers and sellers submit limit orders. If at a given moment in time, the buyer is unable to complete his entire order due to the shortage of sell orders at the required limit price, the unmatched part of the order is recorded in the order book. Subsequently these buy unmatched …
Sharp bounds for first p-Steklov eigenvalues in convex domains.
problem Finding upper bounds for the first p-Steklov eigenvalues in convex domains. method Proved sharp inequalities using isoperimetric and p-Laplacian properties. result Explicit upper bounds for the first p-Steklov eigenvalues in convex domains. Paper derives trace formula for magnetic Laplacian at zero energy.
problem Trace formula for magnetic Laplacian at zero energy.
method Generalizes Gutzwiller trace formula, focuses on zero energy level.
result Derives trace formula at zero energy level.
Formula calculates volume of two-bridge knots.
problem Calculating the volume of two-bridge knots.
method Derived from Hopf formula and Fox derivatives.
result Closed formula for the volume of two-bridge knots.
It has been shown that the Alvarez-Gaumeˊ-Witten miraculous anomaly cancellation formula in type IIB superstring theory and its various generalizations can be derived from modularity of certain characteristic forms. In this paper, we show that the Green-Schwarz formula and the Schwarz-Witten formula i…
Derives integral formulae on weighted manifolds.
problem No specific problem stated; focuses on mathematical derivations.
method Introduces weighted mean sigma-r curvature and uses weighted Newton transformations.
result Derives integral formulae generalizing previous work.
QTAML models quantum tunneling errors for AI robustness.
problem Quantum tunneling errors in AI inference.
method Derives weight-error distribution using WKB approximation, introduces TAC algorithm.
result TAC achieves 95% clean accuracy with 3.4-33.6x less ECC overhead.
A simple formula approximates AUM fees' cumulative costs.
problem Estimating the total cost of AUM fees over time.
method Intuitive explanation and analytical derivation of a formula.
result Investments lose almost Nε% of their value over N years with an annual fee of ε%.
Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas have been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to pres…
The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's…
We derive Verlinde's formula from the fixed point formula for loop groups proved in the companion paper "A fixed point formula for loop group actions", and extend it to compact, connected groups that are not necessarily simply-connected.
The paper derives new Gauss-Bonnet formulas for frontal bundles over surfaces with boundary.
problem Deriving new formulas for coherent tangent bundles over surfaces with boundary.
method Defining frontal bundles and applying Gauss-Bonnet theorems to derive formulas.
result Four new Gauss-Bonnet type formulas for frontal bundles are derived.
Developed new Crofton formulas for pseudo-Riemannian spaces.
problem Computing volumes and curvature integrals in pseudo-Riemannian space forms.
method Introduced Crofton formulas using distributions and Alesker's Radon transform.
result Explicit Crofton formulas for all isometry-invariant valuations on pseudo-Riemannian spaces.
Study finds formulas for minimal submanifolds using Möbius transformations.
problem Understanding minimal submanifolds in Euclidean space.
method Monotonicity formulas for minimal submanifolds involving Möbius transformations.
result Proved formulas for minimal submanifolds under Möbius transformations.
Unified entropy formula for real, complex, and quaternionic DLNs.
problem Deriving a formula for DLNs over different fields.
method Extending Menon and Yu's formula to complex and quaternionic DLNs.
result Unified entropy formula for DLNs over R, C, and H. New Crofton formulae derived from existing ones.
problem Generalizing Crofton formulae for products.
method Calculations in the ring of normal densities.
result Generalizations of Crofton formulae in terms of mixed Riemannian volume.
The paper is devoted to the problem of finding explicit combinatorial formulae for the Pontryagin classes. We discuss two formulae, the classical Gabrielov-Gelfand-Losik formula based on investigation of configuration spaces and the local combinatorial formula obtained by the author in 2004. The latter formula is based…
Formula connects curvature to volume in special geometric spaces.
problem Deriving formulas for curvature in specific geometric spaces.
method Used strong locality of Laplacian and eigenfunction approximation.
result Proved integral type Gauss-Green formula linking curvature to volume.
Wave trace singularity formula for fibre bundles generalizes Poisson summation.
problem Wave trace singularity formula for fibre bundles.
method Proved a wave trace singularity formula for a family of generalised Laplacians defined by a Riemannian fibre bundle.
result Generalizes Poisson summation formulae for families.
Abstract: Characterizes frontals and wavefronts with formulas.
problem Characterizing frontals and wavefronts.
method Representation formulas for frontals and wavefronts.
result Representation formulas for various types of frontals and wavefronts.
Paper proves a generalized Torres formula for twisted Reidemeister torsion.
problem Alexander polynomial properties for links and sublinks.
method Uses twisted Reidemeister torsion to generalize Torres formula.
result Obtains a second proof of Morifuji's result.
New formula simplifies interior polynomial calculation.
problem Calculating interior polynomial efficiently.
method New recursion formula based on non-expanding sets.
result Clearer combinatorial interpretation of interior polynomial.
Computes tube formulas for valuations in complex space forms.
problem Computing values of valuations on complex space forms.
method Develops tube formulas for valuations in complex space forms and generalizes classical formulas.
result Generalizes classical formulas of Weyl, Gray and others.
Formula derived for zeta functions of 3D foliated systems.
problem Analyzing zeta functions of 3D Riemannian foliated dynamical systems.
method Relating dynamical spectral ξ-functions to zeta functions using the distributional dynamical Lefschetz trace formula. result Proved a regularized determinant formula for zeta functions.
We construct a new representation formula for indefinite improper affine spheres in terms of two para-holomorphic functions and study singularities which appear in this representation formula. As a result, it follows that cuspidal cross caps never appear as the singularities on indefinite improper affine spheres and so…
Derives formulas for differential forms on weighted manifolds.
problem Developing formulas for differential forms on weighted manifolds.
method Derives a Reilly formula and explores its applications.
result Proves a Poincaré-type inequality and obtains new eigenvalue estimates.
We extend Kac-Rice formula to compute expected intersections of random submanifolds.
problem Computing expected intersections of random submanifolds.
method Generalized Kac-Rice formula using measure theory and integration.
result Formula computes expected cardinality of preimages of submanifolds via random maps.
We present a gluing formula for Gromov-Witten invariants in the case of a triple product. This gluing formula is a simple case of a much more general gluing formula proved and stated using exploded manifolds. We present this simple case because it is relatively easy to explain without any knowledge of exploded manifold…