arXiv research

New integration theory on topological spaces, including fractals.

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…

Teaches matrix calculus for machine learning and optimization.

Euler calculus is based on integrating simple functions with respect to the Euler characteristic. This paper makes the case for extending Euler calculus to continuous integrands by integrating with respect to (Gaussian) curvature. This requires a metric but is nevertheless defined within any O-minimal theory. It satisf…

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…

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…

We use pathwise Itô calculus to prove two strictly pathwise versions of the master formula in Fernholz' stochastic portfolio theory. Our first version is set within the framework of Föllmer's pathwise Itô calculus and works for portfolios generated from functions that may depend on the current states of the market port…

New calculus solves boundary value problems for elliptic operators.

Characterizes polyhomogeneous symbols and applies to Heisenberg calculus.

Global calculus for manifolds with boundary, solving evolution problems.

This study introduces computation of option sensitivities (Greeks) using the Malliavin calculus under the assumption that the underlying asset and interest rate both evolve from a stochastic volatility model and a stochastic interest rate model, respectively. Therefore, it integrates the recent developments in the Mall…

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…

Develops a new theory of loss functions for statistical machine learning.

The study characterizes complex structures using calculus of variations.

Derives new orthogonal coordinates for evolving surfaces and curves.

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 $S^3$.

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 on spacetimes for nonlinear differential equations.

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.

Abstract: Tangent categories get a Cartan calculus with scalar multiplication by a commutative ring.

Derives FPDE for equity-linked insurance pricing.

Extends calculus to topological manifolds using generalized functions.

Study Brownian motion on Grassmann manifold using matrix stochastic calculus.

In this paper we connect classical differential geometry with the concepts from geometric calculus. Moreover, we introduce and analyze a more general Laplacian for multivector-valued functions on manifolds. This allows us to formulate a higher codimensional analog of Jacobi`s field equation.

We first generalize the operation of formal exterior differential in the case of finite dimensional fibered manifolds and then we extend it to certain bundles of smooth maps. In order to characterize the operator order of some morphisms between our bundles of smooth maps, we introduce the concept of fiberwise $(k,r)$-j…

Develops graphical calculus for monoidal categories with twisted pivotal structures.

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 …

The covariant phase space of a Lagrangian field theory is the solution space of the associated Euler-Lagrange equations. It is, in principle, a nice environment for covariant quantization of a Lagrangian field theory. Indeed, it is manifestly covariant and possesses a canonical (functional) "presymplectic structure" w …

Study of nonlinear PDEs using derived geometry and BV formalism.

We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and…

AP-Calculus offers a new framework for causal inference in Bayesian networks.

Study embedding calculus and link invariants using functor calculus.

Rust library solves complex equations on abstract simplicial complexes.

We show for a certain class of operators $A$ and holomorphic functions $f$ that the functional calculus $A\mapsto f(A)$ is holomorphic. Using this result we are able to prove that fractional Laplacians $(1+Δ^g)^p$ depend real analytically on the metric $g$ in suitable Sobolev topologies. As an application we obtain loc…

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…

We give a necessary and sufficient condition for the existence of a local solution of the inverse problem of calculus of variations in terms of the identical vanishing of the variation of a functional on an extended space (with the number of independent variables increased by one), and explain its relation with the cla…

A new method in finance without probabilities or integrals.

An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which a…

The paper provides an efficient method to price path-dependent derivatives using multiscale stochastic volatility models.

Secondary Calculus formalizes PDEs using cohomology, simplifying their study.

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…

Let $X$ be a compact manifold with boundary. Suppose that the boundary is fibred, $φ:\pa X\longrightarrow Y,$ and let $x\in\CI(X)$ be a boundary defining function. This data fixes the space of `fibred cusp' vector fields, consisting of those vector fields $V$ on $X$ satisfying $Vx=O(x^2)$ and which are tangent to the f…

We investigate the differential calculus defined by Ashtekar and Lewandowski on projective limits of manifolds by means of cylindrical smooth functions and compare it with the C^infty calculus proposed by Froehlicher and Kriegl in more general context. For products of connected manifolds, a Boman theorem is proved, sho…

A diagrammatic language for 3D manifolds with boundary.

In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these methods as well as their interplay. This is a succinct survey that hopes to inspire…

Develops symmetric Cartan calculus linking to Patterson-Walker metric.

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…

The paper simplifies calculus for semimartingales using multiplicative compensation.