arXiv research

Reduces path integrals for interacting systems using dependent coordinates.

Geometrically represents path integral reduction Jacobian for interacting systems.

PS-IG improves feature attribution by reducing noise and variance.

Geometrically represents the Jacobian for a mechanical system with symmetry.

The path integral generalization of the Casson invariant as developed by Rozansky and Witten is investigated. The path integral for various three manifolds is explicitly evaluated. A new class of topological observables is introduced that may allow for more effective invariants. Finally it is shown how the dimensional …

Thermodynamic integration (TI) for computing marginal likelihoods is based on an inverse annealing path from the prior to the posterior distribution. In many cases, the resulting estimator suffers from high variability, which particularly stems from the prior regime. When comparing complex models with differences in a …

GH-PID uses guided harmonic paths for efficient SOT with interpretable diagnostics.

We propose a representation of Gaussian processes (GPs) based on powers of the integral operator defined by a kernel function, we call these stochastic processes integral Gaussian processes (IGPs). Sample paths from IGPs are functions contained within the reproducing kernel Hilbert space (RKHS) defined by the kernel fu…

The paper calculates sensitivities for financial derivatives using path weighting methods.

The paper classifies path structures on 3D Lie groups and reduces non-flat ones to Z/2Z-structures.

In a rigorous construction of the path integral for supersymmetric quantum mechanics on a Riemann manifold, based on Bär and Pfäffle's use of piecewise geodesic paths, the kernel of the time evolution operator is the heat kernel for the Laplacian on forms. The path integral is approximated by the integral of a form on …

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. We compute the Wilson Loop observable using a path integral with an Einstein-Hilbert action. Using axial-gauge fixing, we can write this path integral as the limit of a sequence of Chern-Simons integrals, studied e…

In this paper we analytically study the problem of pricing an arithmetically averaged Asian option in the path integral formalism. By a trick about the Dirac delta function, the measure of the path integral is defined by an effective action functional whose potential term is an exponential function. This path integral …

We study a quantum system in a Riemannian manifold M on which a Lie group G acts isometrically. The path integral on M is decomposed into a family of path integrals on a quotient space Q=M/G and the reduced path integrals are completely classified by irreducible unitary representations of G. It is not necessary to assu…

Researchers calculate the Ray-Singer Torsion for $S^{1}$ bundles.

Path integral method calculates PDBS option prices with time-dependent parameters.

Derives path-integrals for superstrings on curved backgrounds using string geometry theory.

Revisits SYM theory to compute Donaldson invariants using mock modular forms.

In the framework of path integral the evolution operator kernel for the Merton-Garman Hamiltonian is constructed. Based on this kernel option formula is obtained, which generalizes the well-known Black-Scholes result. Possible approximation numerical schemes for path integral calculations are proposed.

We give a rigorous construction of the path integral in N=1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler-Jones-Petrack. Via…

The paper proves signatures of non-geometric rough paths can approximate functionals uniformly.

A new formula connects supersymmetric path integrals to Chern-Simons theory.

Proposes Geodesic Integrated Gradients (GIG) for more accurate feature attributions in deep networks.

Improved path integral method for financial derivatives pricing.

Derives path integrals for perturbative strings on various backgrounds.

The paper analyzes how grid cells perform path integration and learns hexagon grid patterns.

Path integral method calculates barrier option prices.

Many introductory courses in quantum mechanics include Feynman's time-slicing definition of the path integral, with a complete derivation of the propagator in the simplest of cases. However, attempts to generalize this, for instance to non-quadratic potentials, encounter formidable analytic issues in showing the succes…

We give a pragmatic/pedagogical discussion of using Euclidean path integral in asset pricing. We then illustrate the path integral approach on short-rate models. By understanding the change of path integral measure in the Vasicek/Hull-White model, we can apply the same techniques to "less-tractable" models such as the …

We present two different approaches to stochastic integration in frictionless model free financial mathematics. The first one is in the spirit of Itô's integral and based on a certain topology which is induced by the outer measure corresponding to the minimal superhedging price. The second one is based on the controlle…

Foundation for robust finance using rough path theory.

Paper explores rough path theory for frictionless markets, linking NCFL to unbiased rough integrators.

For every adapted, càglàd process (strategy) $G$ and typical càdlàg price paths whose jumps satisfy some mild growth condition we define integral $G\cdot S$ as a limit of simple integrals.

In this paper, we aim to understand Residual Network (ResNet) in a scientifically sound way by providing a bridge between ResNet and Feynman path integral. In particular, we prove that the effect of residual block is equivalent to partial differential equation, and the ResNet transforming process can be equivalently co…

Convolution operations designed for graph-structured data usually utilize the graph Laplacian, which can be seen as message passing between the adjacent neighbors through a generic random walk. In this paper, we propose PAN, a new graph convolution framework that involves every path linking the message sender and recei…

We study finite-dimensional integrals in a way that elucidates the mathematical meaning behind the formal manipulations of path integrals occurring in quantum field theory. This involves a proper understanding of how Wick's theorem allows one to evaluate integrals perturbatively, i.e., as a series expansion in a formal…

Chen's iterated integrals are treated within synthetic differential geometry. The main result is that iterated integrals produce a subcomplex of the de Rham complex on the free path space as well as based path spaces.

We propose a general strategy to derive null-homotopy operators for differential complexes based on the Bernstein-Gelfand-Gelfand (BGG) construction and properties of the de Rham complex. Focusing on the elasticity complex, we derive path integral operators $\mathscr{P}$ for elasticity satisfying $\mathscr{D}\mathscr{P…

We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's pri…

Let M be a compact Riemannian manifold without boundary and let H be a self-adjoint generalized Laplace operator acting on sections in a bundle over M. We give a path integral formula for the solution to the corresponding heat equation. This is based on approximating path space by finite dimensional spaces of geodesic …

Study path geometries with constant torsion and cone structures.

Develops path integral for spiked tensor model dynamics.

We introduce a model for the dynamics of stock prices based on a non quadratic path integral. The model is a generalization of Ilinski's path integral model, more precisely we choose a different action, which can be tuned to different time scales. The result is a model with a very small number of parameters that provid…

In this paper I develop a new computational method for pricing path dependent options. Using the path integral representation of the option price, I show that in general it is possible to perform analytically a partial averaging over the underlying risk-neutral diffusion process. This result greatly eases the computati…

Global invariant for path structures and differential equations defined on torus.

Paper identifies reductive MDPs, solving them in polynomial time.

This is a survey based on joint work with Florian Hanisch and Batu Güneysu reporting on a rigorous construction of the supersymmetric path integral associated to compact spin manifolds.

We provide an introduction to the signature method, focusing on its theoretical properties and machine learning applications. Our presentation is divided into two parts. In the first part, we present the definition and fundamental properties of the signature of a path. The signature is a sequence of numbers associated …