In this paper we consider the problem of the quantile hedging from the point of view of a better informed agent acting on the market. The additional knowledge of the agent is modelled by a filtration initially enlarged by some random variable. By using equivalent martingale measures introduced in Amendinger (2000) and …
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Optimal transport reformulates multiple quantile hedging problem.
problem Multiple quantile hedging problem in incomplete markets.
method Reformulated as Monge optimal transport problem, introduced Kantorovitch version, proved no duality gap.
result Multiple quantile hedging problem can be seen as semi-discrete optimal transport problem.
Develops a new model-free approach to portfolio theory using rough paths.
problem Handles more general portfolios without probabilistic assumptions.
method Rough path theory for stochastic portfolio theory (SPT).
result Asymptotic growth rates of various portfolios match.
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.
Lie Calculusmath.GR
Lie Calculus connects differential and Lie theory using groupoids.
problem Understanding the relationship between differential and Lie theories.
method Using groupoids to link differential and Lie theories.
result Higher order theory involves higher algebra (n-fold groupoids).
N-Cartan calculusmath.RT
New calculus extends noncommutative geometry to higher N.
problem Developing a new noncommutative differential geometry.
method Examining N-Koszul calculus for N-symmetric algebras.
result Extension of N-Cartan calculus to manifolds conjectured.
Study embedding calculus and link invariants using functor calculus.
problem Detect Milnor invariants using embedding towers of string links.
method Use functor calculus and Goodwillie-Weiss embedding calculus.
result Embedding tower detects Milnor invariants.
Embedding calculus proves convergence for surfaces.
problem Proving convergence of embedding calculus for surfaces.
method Goodwillie-Weiss' embedding calculus for spaces of embeddings into a manifold of dimension at most two.
result Relates Johnson filtration of mapping class group to embedding calculus.
Study geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.
problem Geometric characterization of asymptotic pseudodifferential calculus on spinor bundles.
method Groupoid approach to pseudodifferential calculus, rescaled bundle.
result Rescaled bundle provides geometric characterization to asymptotic pseudodifferential calculus on spinor bundles.
Secondary Calculus formalizes PDEs using cohomology, simplifying their study.
problem Formalizing and simplifying the study of partial differential equations (PDEs).
method Using cohomology of diffieties to formalize PDEs and their properties.
result Differential calculus on PDE solution spaces is homotopy calculus on horizontal De Rham algebras of diffieties.
In arXiv:1207.0332 [cs.LO] was proposed a graphic lambda calculus formalism, which has sectors corresponding to untyped lambda calculus and emergent algebras. Here we explore the sector covering knot diagrams, which are constructed as macros over the graphic lambda calculus.
Extends differential calculus to triole algebras.
problem No specific problem stated; focuses on extending differential calculus.
method Generalizes diolic differential calculus to triole algebras with fiber metrics.
result Established a conceptual framework for calculus on bundles with vector-valued fiber metrics.
Unified Lie structures in homotopy and isotopy calculus.
problem Compatibility of Lie structures in homotopy and isotopy calculus.
method New technical tool: bracket on total homotopy fibres of collapsing cubes of wedge sums.
result Unified understanding of Lie structures in homotopy and isotopy calculus.
A diagrammatic language for 3D manifolds with boundary.
problem Representing and manipulating 3D manifolds with boundary.
method Diagrammatic calculus and local moves.
result Completeness of the diagrammatic calculus proved.
Notes on Tractor Calculusmath.DG
Introduces tractors for basic examples and modern differential calculus.
problem None explicitly stated, focuses on introduction.
method Classical examples and modern invariant differential calculus.
result Introduction to tractors and related modern differential calculus.
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…
This memoir deals with the hypoelliptic calculus on Heisenberg manifolds, or Heisenberg calculus. The Heisenberg manifolds generalize CR and contact manifolds and in this context the main differential operators at stake include the Hörmander's sum of squares, the Kohn Laplacian, the horizontal sublaplacian and its conf…
Simplified calculus for semimartingales makes complex transformations easier.
problem Complex transformations of semimartingales.
method Unified treatment of transformations for real and complex semimartingales.
result Unified calculus for semimartingales simplifies various transformations.
Develops symmetric Cartan calculus linking to Patterson-Walker metric.
problem No specific problem stated; focuses on developing a new calculus.
method Symmetric Cartan calculus, using torsion-free affine connections.
result Symmetric Cartan calculus is a complete analogue of classical Cartan calculus.
Operational guide to discrete exterior calculus on cubic cells.
problem Applying calculus on discrete manifolds.
method Defining discrete exterior calculus on cubic cells for discrete manifolds.
result Gauss and Stokes theorems are recovered on the discrete torus.
This is a short description of graphic lambda calculus, with special emphasis on a duality suggested by the two different appearances of knot diagrams, in lambda calculus and emergent algebra sectors of the graphic lambda calculus respectively. This duality leads to the introduction of the dual of the graphic beta move…
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.
Accessible proof of Gauss-Bonnet theorem using vector calculus.
problem Proving the Gauss-Bonnet theorem for surfaces in 3D.
method Classical vector calculus approach.
result Self-contained proof of Gauss-Bonnet theorem.
New algebraic formalism for differential calculus in Diolic algebras.
problem Studying differential calculus in vector bundles.
method Introducing functors of differential calculus over arbitrary graded commutative algebras (DCGCA) and applying this to Diolic algebras.
result Recovery of well-known objects and notions from ordinary differential, symplectic, and Poisson geometry, with unique aspects.
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.
Symbol calculus extended for foliations' transverse geometry.
problem Understanding index theory of transversely elliptic operators on foliations.
method Constructing Getzler rescaling calculus and Block-Fox calculus of asymptotic operators.
result Composition of ADOs is again an ADO, with a leading symbol formula.
Introduces scale calculus and M-polyfolds for graduate students.
problem No specific problem stated in the abstract.
method Lecture notes on scale calculus and M-polyfolds.
result No specific key result mentioned in the abstract.
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.
Following the programme set out in Part I of this work, we develop a conceptual higher order differential calculus. The '' local linear algebra '' defined in Part I is generalized by '' higher order local linear algebra ''. The underlying combinatorial object of such higher algebra is the natural n-dimensional hyper-cu…
Develops calculus on a new type of supermanifolds.
problem No new problem introduced, just generalization of supergeometry.
method Developed differential calculus on $\Zn$-supermanifolds.
result Foundations of differential calculus established on $\Zn$-supermanifolds.
Cartan calculus applied to string topology homology.
problem Understanding the structure of free loop spaces.
method Introduced Cartan calculus on loop homology, linked to string topology operations.
result Loop product and bracket behavior under Hodge decomposition.
New method constructs deformation groupoid for inhomogeneous pseudo-differential calculus.
problem Recovering inhomogeneous pseudo-differential calculus using a deformation groupoid.
method Elementary construction of deformation groupoid for Heisenberg calculus, then generalization to arbitrary filtrations.
result Elementary construction of deformation groupoid for inhomogeneous pseudo-differential calculus.
Develops a graphical calculus for stable curvature invariants.
problem Calculating stable curvature invariants of Riemannian manifolds.
method Graphical calculus based on trivalent graphs with colored edges.
result Derives a curvature identity for compact Einstein manifolds.
Geometric calculus introduced on pseudo-Riemannian manifolds without embedding.
problem Developing calculus on pseudo-Riemannian manifolds without embedding.
method Direct axiomatic approach to geometric calculus, paralleling general relativity.
result Full theory of differential calculus for vector, multivector, and tensor fields developed.
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.
Coarse pseudo-differential calculus and index theory on manifolds with a tangent Lie structuremath.DG
Simplified calculus for manifold operators, proving index theorems.
problem Developing calculus for manifold operators and proving index theorems.
method Introducing a simplified pseudo-differential calculus for zero-order operators on manifolds with a tangent Lie structure.
result Proving index theorems for `h-elliptic' operators on manifolds with a tangent Lie structure.
Derives optimal control conditions using calculus of variations.
problem Optimizing Markov control in stochastic control problems.
method Calculus of variations approach to derive necessary conditions.
result Solves the Merton portfolio optimization problem.
The calculus correspondence has been known to exist between generic pedal evolutions and generic wave front evolutions. In this paper, we first extend the known results on the calculus correspondence to evolutions with multi-parameters, and then give applications of calculus correspondence. Moreover, we discuss the pos…
Extends calculus theorem to higher dimensions.
problem Calculus theorem limitations in higher dimensions.
method Type Stokes' theorem for type -chains.
result Extends fundamental theorem of calculus.
To give a Cartan calculus on the extended quantum 3d space, the noncommutative differential calculus on the extended quantum 3d space is extended by introducing inner derivations and Lie derivatives.
Quantum calculus models stock liquidity issues.
problem Capturing illiquidity in stock price distributions.
method Quantum stochastic calculus applied to finance.
result Modeling the impact of widened bid-ask spreads.
Geometric DEC solves Poisson on general triangulations.
problem Solving Poisson equation on arbitrary triangulations.
method Revisited Discrete Exterior Calculus (DEC) for general triangulations using Vector Calculus and Matrix Algebra.
result DEC solutions match FEML for Poisson equation.
Author provides notes on differential calculus on special metric spaces.
problem Developing calculus on complex metric spaces.
method Self-contained overview of recent developments with simplified assumptions.
result Presentation of differential calculus ideas on RCD spaces.
We introduce and study graphic lambda calculus, a visual language which can be used for representing untyped lambda calculus, but it can also be used for computations in emergent algebras or for representing Reidemeister moves of locally planar tangle diagrams.
Develops calculus for tamed Dirichlet spaces using measure theory.
problem Defines calculus for measure spaces with Dirichlet forms.
method Introduces first and second order calculus on tamed Dirichlet spaces.
result Defines various geometric objects on tamed Dirichlet spaces.
Koszul duality for manifold modules proven.
problem Proving Koszul self duality of manifold modules.
method Using generalized Thom complexes and operads in Top.
result Koszul self duality of little disk modules proven.
Unified approach to Finsler geometry connections simplifies calculus of variations.
problem Simplifying the calculus of variations in Finsler geometry.
method Unified approach to connections in sprays and Finsler metrics.
result All classical Finslerian connections yield same calculus of variations formulas.
Simplified Khovanov-Rozansky calculus for bipartite knots.
problem Complexity in calculating superpolynomials for knots.
method Bipartite calculus generalizes Khovanov-Rozansky calculus for a restricted class of knots.
result Simplification of Khovanov-Rozansky polynomials for bipartite knots.