arXiv research

Formula derived for zeta functions of 3D foliated systems.

Extends Itô's formula for path-dependent functions in finance.

We give a new formula for the energy functionals E_k defined by Chen-Tian, and discuss the relations between these functionals. We also apply our formula to give a new proof of the fact that the holomorphic invariants corresponding to the E_k functionals are equal to the Futaki invariant.

The aim of this article is to generalize in several variables some formulae for Eisenstein series in one variable. For example the formula $2ζ(2k) = (2π)^{2k} \frac{B_{2k}}{(2k)!} = Res_{z=0}(\frac{1}{z^{2k}(1-e^z)})$ for the values of zeta functions at even integers in functions of Bernoulli numbers. A. Szenes proved …

We establish a few formulas that compute the volume of the zero-set (or nodal set) of a function on a compact Riemannian manifold as integrals of functionals of the function and its derivatives.

We study a functional that derives from the classical Yang-Mills functional and Born-Infeld theory. We establish its first variation formula and prove the existence of critical points. We also obtain the second variation formula.

Author presents the second variational formula for statistical biharmonic maps.

A new formula reveals symmetries between mean excess and ES functions.

In this paper, we obtain asymptotic formulas with error estimates for the implied volatility associated with a European call pricing function. We show that these formulas imply Lee's moment formulas for the implied volatility and the tail-wing formulas due to Benaim and Friz. In addition, we analyze Pareto-type tails o…

Guillemin trace formula adapted for group actions.

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…

New formula and properties of inverted Habiro series derived from GM series.

Derives functional Itô formula for non-anticipative maps of rough paths.

Extending Itô's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer-Itô, applies to one dimensional semimartingales and convex functions. There are also satisfactory generalizations of Itô's formula for diffusion processes whe…

Derives a new formula for optimal stopping problems with exploding derivatives.

In this paper, we study monotonicity formulas of eigenvalues and entropies along the rescaled List's extended Ricci flow. We derive some monotonicity formulas of eigenvalues of Laplacian which generalize those of Li in [8] and Cao-Hou-Ling in [3]. Moreover, we also consider monotonicity formulas of $\mathcal{F}_k$-func…

A formula connects discrete harmonic surfaces to holomorphic functions.

We consider an example of tubes of hypersurfaces in Euclidean space and generalise the tube formula to supercase. By this we assign to a point of the hypersurface in superspace a rational characteristic function. Does this rational function appear when we calculate the zeta-function of an arithmetic variety?

Derives formulas from Green function Hessian assumption.

The Lax-Hopf formula simplifies the value function of an intertemporal optimization (infinite dimensional) problem associated with a convex transaction-cost function which depends only on the transactions (velocities) of a commodity evolution: it states that the value function is equal to the marginal fonction of a fin…

Proves a formula for Kontsevich-Witten tau-function using Schur Q-polynomials.

Researchers extend monotonicity formulas for harmonic functions in RCD(0,N) spaces.

Study on multiple linking numbers, extending Gauss diagram formulas.

Krein's formula for conic Laplacians on compact Riemann surfaces

Polterovich proved a remarkable closed formula for heat kernel coefficients of the Laplace operator on compact Riemannian manifolds involving powers of Laplacians acting on the distance function. In the case of Kähler manifolds, we prove a combinatorial formula for powers of the complex Laplacian and use it to derive a…

In this paper we prove an approximate formula expressed in terms of elementary functions for the implied volatility in the Heston model. The formula consists of the constant and first order terms in the large maturity expansion of the implied volatility function. The proof is based on saddlepoint methods and classical …

The article proves a new entropy formula for surfaces with boundaries.

Paper solves bond option pricing with credit risk using Black-Scholes equations.

Kernel for STL formulae enables machine learning in temporal logic.

Some expansion methods have been proposed for approximately pricing options which has no exact closed formula. Benhamou et al. (2010) presents the smart expansion method that directly expands the expectation value of payoff function with respect to the volatility of volatility, then uses it to price options in the stoc…

We consider a standard optimal investment problem in a complete financial market driven by a Wiener process and derive an explicit formula for the optimal portfolio process in terms of the vertical derivative from functional It^o calculus. An advantage with this approach compared to the Malliavin calculus approach is t…

The paper studies biharmonic functions and bi-eigenfunctions on spheres and model spaces.

We give reconstruction formulas inverting the geodesic X-ray transform over functions (call it $I_0$) and solenoidal vector fields on surfaces with negative curvature and strictly convex boundary. These formulas generalize the Pestov-Uhlmann formulas in [Pestov-Uhlmann, IMRN '04] (established for simple surfaces) to ca…

The paper investigates quantitative rigidity using Colding's monotonicity formulas for Ricci curvature.

New proof of Positive Mass Theorem using Green's function and monotonicity formula.

Paper introduces cubature method for stochastic Volterra equations.

We establish the equivalence of the Tuynman midpoint area formula for a spherical triangle to the classical area formulas of Euler and of Cagnoli. The derivation also yields a variant of the Cagnoli formula in terms of the medial triangle. We introduce the three barycentric coordinates of a point within the spherical t…

The Lugannani-Rice formula is a saddlepoint approximation method for estimating the tail probability distribution function, which was originally studied for the sum of independent identically distributed random variables. Because of its tractability, the formula is now widely used in practical financial engineering as …

Maximum Levine-Tristram signature of torus knots follows a reduction formula.

The paper establishes a Poisson integral formula for bounded pluriharmonic functions on Teichmüller space.

In this effort, we derive a formula for the integral representation of a shallow neural network with the ReLU activation function. We assume that the outer weighs admit a finite $L_1$-norm with respect to Lebesgue measure on the sphere. For univariate target functions we further provide a closed-form formula for all po…

Study on contact Hamiltonian functions for singular contact structures.

On a constraint manifold we give an explicit formula for the Hessian matrix of a cost function that involves the Hessian matrix of a prolonged function and the Hessian matrices of the constraint functions. We give an explicit formula for the case of the orthogonal group ${\bf O}(n)$ by using only Euclidean coordinates …

We prove optimality of the Arf invariant formula for the generating function of even subgraphs, or, equivalently, the Ising partition function, of a graph.

The study derives formulas for functionals on surface with boundary under harmonic Ricci flow.

Paper proves gluing formula for analytic torsions using Witten deformation for non-Morse functions.

We extend the model-free formula of [Fukasawa 2012] for $\mathbb E[Ψ(X_T)]$, where $X_T=\log S_T/F$ is the log-price of an asset, to functions $Ψ$ of exponential growth. The resulting integral representation is written in terms of normalized implied volatilities. Just as Fukasawa's work provides rigourous ground for Ch…

New IBP formulae for rough stochastic Volterra processes.